Volume 12 (1866) (428 pages)

290 a — wil Seientific Lress. Conmmenications. a gas DEPARTMENT we luvite the FREE DiscussION of a roper alone belng 1 ae licas and theories they advance. \ Nay [Written for the Mining and Scientific Press.] \ THE TRACTORY CURVE, AND THE PROPERTIES OF GRINDING ag Serr 11 ‘ } yt BY .W. ‘pe @QoDyEaR, i if a Civil snd . Miniog a P Fi ‘ Ot thankMi. Rahdall for the compliments he has been pleased” to bestow-upon me; and which form so large a part of his receut witty and classic reply to my first artiele upon the tmctory eurve nnd the properties of.grinding plates; hut ns neither my own scientifie aequirements, nor those of the nuthor of tho Quartz Operator’s Hand Book, constitute . precisely the question which I had proposcd to discuss, he will perhaps exeuse me for lenying this -portion.of his. article unanswered. . a “A will now first aeknowledgo nnd_ correct two, errors into which I haye fallen, and for which my only excuse is the faet that my former article was rapidly written -last January, and that since that date I have not looked at the subject farther than simply to corrcet the “ proof” for the printer, till my attention was again, called to it hy Mr. Rap. dall’s reccut article. . ‘ These errors oceur in that portion of my article relating to conical and curved grinding ‘surfaces, and vitiate, of course, in certain respects the results’ which I obtained for sueh. Tbe first consists in decomposing, as I have done on page 162, the vertieal pressure at ony point of eurved plates into two components, one of which is. normal to the surface, and the other parallel to the tangent of the meridinn curve. This force should be so, decomposed that one of the components is normal to the surface, while the other one, lying i in the plane of the meridian eurve, is horizontal, 7. e., perpendicular to the axis. This ‘horizontal component is then met and neutralized by an equal and‘ opposite force coming from the ‘opposito side of the axis, where n similar deeomposition takes place. ‘Tho second error consists in tacitly nssuming that the vertical pressuro per' mnit of superficial area, npon any portion of tho -curved surface, is cqual to the vertical pressure per unit of area measured upon the projection of that: isurface on a horizontal plane, z:¢., on a plane per‘pendienlar to the axis. Having stated tbeso errors, I will now givo gen-. eral formulas in whieh they are corrected, ond which will he found to yield correet results. uo Let a represent the angle included betweon the exis; and the tangent tothe meridian curve at tho point zy. Lct P, as in my first artiele, represent ‘the vertical pressuro per unit of horizontal arca at ‘tho circumference of the plates. Let P’ represent the vertical pressuré per uuitof horizontal aren nt the point x,y. Let P’’ represent the vertical pressure ser unit of superficial area nt the point x,y. Let P’’* represent the normal pressure per unit of superficial avea'at the same'point. Let s represont,: ‘as before, the length‘of any portion of the meridian leurve. Let [email protected] the: exterior radiuSof the «plates, and let frepresent the co-efficient of frietion. The following relations then exist between these quuntitics :, (1) ea @) 0080, a = oa 7a, * ad i em 3 p! — ae ey 4)., Ny — [pS ae er (5) . Fs = the value; which P’ assumes when y =a. %For surfaces whose exterior radius is n, the following formulas are then general : (8) Total vertical pressure, 7 afer hPY y Vf oss ii ; P! y ay (7) Total normal pressure, =2n Py Ba ag " 6) ‘otal foree of friction, rz ee a = = 2 FP'y S ait (9) Moment ofaftiedeny = =f 2 AFP’ y? a, 410) Meahitiical effect, of fri a, ina single reyoai 72 Sadt pr afi aePly (11) Total _ effect in a aie rov guntion, . ds dy. ay ¥ y For complete or entiro surfaces, the limits beee" ha = f4 Py 4 tween whieh theso integrals must be taken, are 0 and a, and for ring surfaces, the inner radius nnd tbe onter-radius. In what follows I shall assume that P is known, und is the same in all cases. Now, when I use the phrase “‘ vertical wear,” I mean by it. the thickness measured in a vertieal direttion of the layer of material worn from the plates by n given omount of motion, ond I still maintain, and for the samo reasons as in my first article, that uniformity of verticnl wear is n condition thnt does not depend npon the particular form of tho grinding surfaee, hut is one of the necessary conditions of grinding with solid plates of nny form. If this bo admitted (and it scems so simple as to be almost an axiom), it follows at once thnt with flat plates of uniform hardness, P’ must vary inversely as y, and that in order that P’ may he constant, tho hardness of the plates must vary as y, and in this artielo T shall nssume that the same law holds for curved surfaees. Weishach, now, expressly states that he assumes the vertical pressure to be equally distributed. The eases then whieh ho considers are those in which tho hardness varies as y. For these cases P* is eonstant and equals P, and with this value of P’ it will be found that formulas (6), (9), and (10), as given above, agree with all of his results. The relative, grinding effects, then, fora siugle revolution with entire plates, whoso harduess varies as the ordinate, are as follows : (12) for flateplaics ie oe. oe. »6667 (18) for conical plates... 7453 (14) for tractory conoidal plates. .1,0000 tho hight of the cone in (18) being A == > and the weight of the plates being tho same in all three cases. These results are given by formula (11). If, on the other hand, the plato be of uniform hardness, then P* vnries inversely as y, and is cqual to PP. Taking this valuo of P*, we obtain from y formula (11) tbe grinding effeet in one revolution : (15) for flat plates...207 Pn’. (16) for conicnl plates.20? Pa \/a® +22. (17) for tractory conoidal plates.4 1? Pa? . Or, velntively, taking as I have dono in (12), (13), and (14), 2n? Pa? ns tho unit, (18) for flat plates (19) for conical plates (20) for traetory conoidal plates.2,000 . 1,000 the weight of the plates in these tbreo cases being the samo. The pressure P né the circumference is tho same in all six of the nhove cases; but in the last three, where the hardness is uniform, the totnl pressure or weight of tbo plates is twice as great as it is in tbe first tbree eases where the hrdness varies as y. If wo assume the weight of tho plates to be the same in nll six cases, the pressure at the cireumference will be only half as great in the last tbree cases as in the first, and tbo ratio of grinding cffect will ‘stand as follows : For plates whose hardness varies as y— (21) flat plates.. ES ase +. ,6667 * (22) conical plntes. . . 57458 : (23) tractory cater ater -1,0000 "For plates of uniform hardness— ,; (24) flat plates... APNE «50:0 »5000 (25) conical plates...+. ,5590 {26) tractory conoidal plates. .1,0000 For ring plotes whose outer radius is n, and avhose inner radius is ; » and whose total weight is the same ns that of the entire plates just gorsiddered, these ratios will stand : For plutes whose hardness varies as y— ‘! (27)Pilatiplatetemern 0. 71222. (28) conical Muites, oem. . ceBIC 6 ,8075 (29) tractory conoidal plates. . 1 10000" For plates of uniform hardness— (30) flat plates..8.0+00++,6667 (31) conieal plates.... 57453 (82) traetory conoidal piutes MN 1,0000 1 and for— asics . ” : (33) Rondall’s Patent Grinding EBLILES ere. ee 37300 In my first articlo (page 210) I gave certain ratios for the grinding effect of ring plates, snpposi} ing the vertical pressure at tbe eirenmferenco to be the same in all cases, the total weights of the plates being different of course. Thoso ratios for the flat plates were correct-under that supposition, though, ns I have alrendy stated, they were ineorreet for the conical and tractory conoidal plates. The result there stated for Randali’s Plates is also incorrect, for the following reason hh, the calculation hy which I obtained it _on~page 178, T-committed one of Mr. Randall’s “school-b vitor) in assuming thnt the pressure at any point of the, surfaco of the inner ring was equal to the pressure at its onter circumference, multiplied bys zl user Sy forme is, in faet, equal to the latter’ saultigigd . by 22 ay’ and with this correction, the resuilt which I gnve as 1,4444, will he found to Ye 1,3333. Le corrections for the other cascs are simple, and are. given dircetly by formula (11). I have nowexplained and corrected my errors. Mr. Randall eomplains that in my first article I did not tell the wholo truth, inasmuch as I did not. state comparative results for plates of equal weight, instead of for plates having equal vertical pressures at the circumference. Ihave now, therefore, conformed myself to his standard in this respeet, and the results in this article from (21) to (83) inelusiye: . are for plates whose total weight is iu all cases the’ same, viz.,1 Pn?. The vertical preseures per nuit of horizontal aren at tho circumference, iu. pied cases, are as follows For (21), (22) and (23)...0.P. For (24), (25) und (26).. a ' PB For (27), (28) and (29)... 4 P. For (30), (31) and (82).. _ : Foye (3) caesar bemecerrarcs ctor a: By eomparing the above results with Mr. Randall’s recapitulation, on page 260, it will be seen tbat we now agree with reference, first, to tho mu, tual ratios for entire plates of equal weight nmong themselves, and second, to the mutual ratios for ring plates of equal weight nmong themselves, but that we still disagree as to the ratio between entire plates . and ring plates of the same weight. The reason of this is.that in Mr. Randall’s discussion, whether he knows it or not, he hns not taken the conditions such that the total weight of his ring plates js the same as thnt of his entire plates; hut he has taken his ring plates sueh that their total weight is equal to tho weight of that portion of his cntiro plates which would be left if a circular portion whose’ radius is wero removed from their center. Tho total weight of his ring plates is, therefore, equal to ; of that of his ontire plates; and if his numhers for ring plntes he all multiplied by z tho corresponding plates will then all havo the samo weight, and our numerical results will agrce tbroughout. I had supposed that in my first article I had at . : Icast made my meaning so clear thnt a mathematician could not well misunderstand it; but I find that Mr. Randnll has done so, and there are in his . article some flagrant misrepresentations that I cannot allow to pass uunotieed. In speaking of the secondary eurve which I discussed, ho says: “Mr. Goodyear errs in premising . ’ that the wear to the plates takes place” (the itnlies are mine) ‘in the direction of the normal reekoned from the point of tangeney.’”’ I premised no such thing. By referring to page 162, he will see that I use tho words, “whero this wear would actually take place, if it could tnke place nt. all,” and the wholo tenor of my articlo went to show that such a wear could noé tako plaeo at oll, I discussed tho seeondary eurve, because I thoaght it involved in the proposition on page 103 of the Hand Book. If Ihave misunderstood tbo author upon this point, I regret it; hut I submit that tho man of straw was not entirely of my own ereating, and that the proposition referred to, if it means anything as there stated, may justly be taken to in. yolve the idea of this very wear which cannot take place, and, consequently, of ‘the “secondary, eurye which I discussed. " Again, on page 259, Mr. Randall says “ 1 Pa? is tho volume or size of the plate whieh Mr. Goodyear tells the pnblie has for its grinding effect 29° Pn* .” I did not tell them this; but I told them (and tho stntement is true) that, with circular plates of equal radius, aud having equal vertical pressures (P) nt the circumference, the grindiug effeet of plates,’ whose, hardness is uniform, is 22 Pa}, while if the hardness vary as y, it is 4 =n? Pa?. I distinetly stated, too, (and Mr. Randull has quoted the statement, with_an nceompnnying question that seems to me absurd, considering tho clenmess with. whieh«I stnted the standard of eomparison which I was using,) that in these two cases the otal pressure or weight (corresponding to what Mr, Randall here calls the “ volume”) was greater in the ono case than in the other. It is, in faet, twico as great, Also, on page 260, Mr. Randall says: “ Mr. Goodyear wodld show by his (fgures)-that a plane plate of uniform hardness will grind fifty per cent. more tlian a plate of the same size and, werght2 (the, italies are mine) “ and hardness‘at the eircamference, and which deeresges ‘ius hardness toward, athe eénter ‘in thé ratio of . the deereuse of ‘the radius.” ‘This is not what I said or dttempted to show. It would not be true. What I did say-will be apparent by referring to my article, and I was viele upon this points 4; On the same page lie says: “ z= tells us that the entire plate of any of the forms eonsidered will grind more than the corresponding ring plate of the proportions given, which I have proven by the authorities and demonstration to hen misstatement. He tells os that the plane plate of uniform hardness will grind more than Randall’s Patent Grindivg .Plates, which statement is doubly refuted.” Two more misrepresentations of the same kitid! I mado no such statements lor ‘plates olvequal weight. I made them only for. plates having equal pressures at ‘the eircumference, nnd for such plates hoth statements are trae, My comparisons throughout that articlo were for equal pressures at the circumference, and not for oqual weights of the plates, and I said o then. Mr. Randall's solntion of the problem of plane and conical grinsling plates, on page 260, if I understand it, tacitly admits the principle of the distribution of pressure which I have Fennnciated; for otherwise, his. proportion, “S$: .V:: EB: G” could uot hold trae for solid plates. He is approaching-tho truth.He has elready corrected various-errors in the Hand Book relating to ring formed plates; and when in his calculations and statements of comparative results he makes his ring plates really of the sume weights his eutire plates, he will he bearer right than he has been yet. If. it. would oof he saying too much, however, I would snggest that his demonstrations would be ug clear and no less elegant if he would not confound volume with pressure. The yolome of a solid of uniform density is, of course, proportional to its weight and to the total pressure: produced by that weight ; still, when dealing in fact. with pressure instead of . volume, it 1g just as easy to say “ pressure,” and in some cases it is less likely to lead to indistinct and, possibly, erroneots ideas, His misunderstandings and misrepresentations of my clearlyexpressed meaning evidently arose, witlf one exception, sdlely from his confusion . of the ideas of volume and pressure. ., ’ “It will be seen from what proceeds that I now admit that differently-formed plates of equal radii, hardness, and weight produce very different amounts of ‘grinding effect inn singlo revolution. ‘The tractory convid, however, ‘by no means gives n maximani of grinding effect nuder these conditions. ‘Ihe expression for thé grinding effect of the cone isa function of Phy the hight of the cone, and by increasing h, we can inerease without limit the theoretical . grinding effect, while still preserving the same . weight, eté., of the plate. But there is another point upon which I must say a few words, though Mr. Randall does not wish it “ logged into ” the question, ‘aud that is power or mechaniéal effect. ‘Lhe point which I had proposed to discuss was not simply the relative amount of grinding effect produéed by plates of dilferent forms. It was the broader qaestion of the relative advantages in the grinding effect to be derived from differentiy-lormed grinding surfaces, and into this question the element of power neces‘sarily enters. “It will he seen that formula (10), which gives ‘the mechanical effect or power eonsumed by the friction:in:a single revolution, and formula 11), which gives the grinding effect, pre identical, with the exception of. the. coelficient of frietion f, which is constant, and as there formulas nore geueral, the: law is also generul, whatever bo the form of the grinding surface, that the mechanical effect: or power expended is directly proporlional to the grinding effect produced. Mr. Randall, on pasé'260, after first objecting to any consideration: of the: element of power, says: “ No mention iu that work (the Hand Book) is made of power in connection with grinding effect. It requires no demonstration to prove that it takes more pozer to reduce ten tons of rock than it does to reduee one to the samo degree of fineness from the same condition at first. Any school-boy can tell us that, saying nothing of practiced miners and mechanics. What*was claimed, and is claimed, and is true, is that the tractory formed pan will do very mach more work than cither the flat or conical pan-of ithe same weight, andi I will uow add do the samo anlount of work at a less expenso of power.” The Hand Book, indeed, makes no mention of power, und, therefor 2; (in addition to its other errors,) the whole character of its dis

290 a — wil Seientific Lress. Conmmenications. a gas DEPARTMENT we luvite the FREE DiscussION of a roper alone belng 1 ae licas and theories they advance. \ Nay [Written for the Mining and Scientific Press.] \ THE TRACTORY CURVE, AND THE PROPERTIES OF GRINDING ag Serr 11 ‘ } yt BY .W. ‘pe @QoDyEaR, i if a Civil snd . Miniog a P Fi ‘ Ot thankMi. Rahdall for the compliments he has been pleased” to bestow-upon me; and which form so large a part of his receut witty and classic reply to my first artiele upon the tmctory eurve nnd the properties of.grinding plates; hut ns neither my own scientifie aequirements, nor those of the nuthor of tho Quartz Operator’s Hand Book, constitute . precisely the question which I had proposcd to discuss, he will perhaps exeuse me for lenying this -portion.of his. article unanswered. . a “A will now first aeknowledgo nnd_ correct two, errors into which I haye fallen, and for which my only excuse is the faet that my former article was rapidly written -last January, and that since that date I have not looked at the subject farther than simply to corrcet the “ proof” for the printer, till my attention was again, called to it hy Mr. Rap. dall’s reccut article. . ‘ These errors oceur in that portion of my article relating to conical and curved grinding ‘surfaces, and vitiate, of course, in certain respects the results’ which I obtained for sueh. Tbe first consists in decomposing, as I have done on page 162, the vertieal pressure at ony point of eurved plates into two components, one of which is. normal to the surface, and the other parallel to the tangent of the meridinn curve. This force should be so, decomposed that one of the components is normal to the surface, while the other one, lying i in the plane of the meridian eurve, is horizontal, 7. e., perpendicular to the axis. This ‘horizontal component is then met and neutralized by an equal and‘ opposite force coming from the ‘opposito side of the axis, where n similar deeomposition takes place. ‘Tho second error consists in tacitly nssuming that the vertical pressuro per' mnit of superficial area, npon any portion of tho -curved surface, is cqual to the vertical pressure per unit of area measured upon the projection of that: isurface on a horizontal plane, z:¢., on a plane per‘pendienlar to the axis. Having stated tbeso errors, I will now givo gen-. eral formulas in whieh they are corrected, ond which will he found to yield correet results. uo Let a represent the angle included betweon the exis; and the tangent tothe meridian curve at tho point zy. Lct P, as in my first artiele, represent ‘the vertical pressuro per unit of horizontal arca at ‘tho circumference of the plates. Let P’ represent the vertical pressuré per uuitof horizontal aren nt the point x,y. Let P’’ represent the vertical pressure ser unit of superficial area nt the point x,y. Let P’’* represent the normal pressure per unit of superficial avea'at the same'point. Let s represont,: ‘as before, the length‘of any portion of the meridian leurve. Let [email protected] the: exterior radiuSof the «plates, and let frepresent the co-efficient of frietion. The following relations then exist between these quuntitics :, (1) ea @) 0080, a = oa 7a, * ad i em 3 p! — ae ey 4)., Ny — [pS ae er (5) . Fs = the value; which P’ assumes when y =a. %For surfaces whose exterior radius is n, the following formulas are then general : (8) Total vertical pressure, 7 afer hPY y Vf oss ii ; P! y ay (7) Total normal pressure, =2n Py Ba ag " 6) ‘otal foree of friction, rz ee a = = 2 FP'y S ait (9) Moment ofaftiedeny = =f 2 AFP’ y? a, 410) Meahitiical effect, of fri a, ina single reyoai 72 Sadt pr afi aePly (11) Total _ effect in a aie rov guntion, . ds dy. ay ¥ y For complete or entiro surfaces, the limits beee" ha = f4 Py 4 tween whieh theso integrals must be taken, are 0 and a, and for ring surfaces, the inner radius nnd tbe onter-radius. In what follows I shall assume that P is known, und is the same in all cases. Now, when I use the phrase “‘ vertical wear,” I mean by it. the thickness measured in a vertieal direttion of the layer of material worn from the plates by n given omount of motion, ond I still maintain, and for the samo reasons as in my first article, that uniformity of verticnl wear is n condition thnt does not depend npon the particular form of tho grinding surfaee, hut is one of the necessary conditions of grinding with solid plates of nny form. If this bo admitted (and it scems so simple as to be almost an axiom), it follows at once thnt with flat plates of uniform hardness, P’ must vary inversely as y, and that in order that P’ may he constant, tho hardness of the plates must vary as y, and in this artielo T shall nssume that the same law holds for curved surfaees. Weishach, now, expressly states that he assumes the vertical pressure to be equally distributed. The eases then whieh ho considers are those in which tho hardness varies as y. For these cases P* is eonstant and equals P, and with this value of P’ it will be found that formulas (6), (9), and (10), as given above, agree with all of his results. The relative, grinding effects, then, fora siugle revolution with entire plates, whoso harduess varies as the ordinate, are as follows : (12) for flateplaics ie oe. oe. »6667 (18) for conical plates..... 7453 (14) for tractory conoidal plates. .1,0000 tho hight of the cone in (18) being A == > and the weight of the plates being tho same in all three cases. These results are given by formula (11). If, on the other hand, the plato be of uniform hardness, then P* vnries inversely as y, and is cqual to PP. Taking this valuo of P*, we obtain from y formula (11) tbe grinding effeet in one revolution : (15) for flat plates......207 Pn’. (16) for conicnl plates..20? Pa \/a® +22. (17) for tractory conoidal plates.4 1? Pa? . Or, velntively, taking as I have dono in (12), (13), and (14), 2n? Pa? ns tho unit, (18) for flat plates (19) for conical plates (20) for traetory conoidal plates..2,000 . 1,000 the weight of the plates in these tbreo cases being the samo. The pressure P né the circumference is tho same in all six of the nhove cases; but in the last three, where the hardness is uniform, the totnl pressure or weight of tbo plates is twice as great as it is in tbe first tbree eases where the hrdness varies as y. If wo assume the weight of tho plates to be the same in nll six cases, the pressure at the cireumference will be only half as great in the last tbree cases as in the first, and tbo ratio of grinding cffect will ‘stand as follows : For plates whose hardness varies as y— (21) flat plates.... ES ase +. ,6667 * (22) conical plntes. . . 57458 : (23) tractory cater ater -1,0000 "For plates of uniform hardness— ,; (24) flat plates..... APNE «50:0 »5000 (25) conical plates......+. ,5590 {26) tractory conoidal plates.. .1,0000 For ring plotes whose outer radius is n, and avhose inner radius is ; » and whose total weight is the same ns that of the entire plates just gorsiddered, these ratios will stand : For plutes whose hardness varies as y— ‘! (27)Pilatiplatetemern 0.. 71222. (28) conical Muites, oem. . ceBIC 6 ,8075 (29) tractory conoidal plates.. . 1 10000" For plates of uniform hardness— (30) flat plates....8.0+00++,6667 (31) conieal plates....... 57453 (82) traetory conoidal piutes MN 1,0000 1 and for— asics . ” : (33) Rondall’s Patent Grinding EBLILES ere. ee 37300 In my first articlo (page 210) I gave certain ratios for the grinding effect of ring plates, snpposi} ing the vertical pressure at tbe eirenmferenco to be the same in all cases, the total weights of the plates being different of course. Thoso ratios for the flat plates were correct-under that supposition, though, ns I have alrendy stated, they were ineorreet for the conical and tractory conoidal plates. The result there stated for Randali’s Plates is also incorrect, for the following reason hh, the calculation hy which I obtained it _on~page 178, T-committed one of Mr. Randall’s “school-b vitor)

in assuming thnt the pressure at any point of the, surfaco of the inner ring was equal to the pressure at its onter circumference, multiplied bys zl user Sy forme is, in faet, equal to the latter’ saultigigd . by 22 ay’ and with this correction, the resuilt which I gnve as 1,4444, will he found to Ye 1,3333. Le corrections for the other cascs are simple, and are. given dircetly by formula (11). I have nowexplained and corrected my errors. Mr. Randall eomplains that in my first article I did not tell the wholo truth, inasmuch as I did not. state comparative results for plates of equal weight, instead of for plates having equal vertical pressures at the circumference. Ihave now, therefore, conformed myself to his standard in this respeet, and the results in this article from (21) to (83) inelusiye: . are for plates whose total weight is iu all cases the’ same, viz.,1 Pn?. The vertical preseures per nuit of horizontal aren at tho circumference, iu. pied cases, are as follows For (21), (22) and (23).....0.P. For (24), (25) und (26).... a ' PB For (27), (28) and (29)..... 4 P. For (30), (31) and (82)... _ : Foye (3) caesar bemecerrarcs ctor a: By eomparing the above results with Mr. Randall’s recapitulation, on page 260, it will be seen tbat we now agree with reference, first, to tho mu, tual ratios for entire plates of equal weight nmong themselves, and second, to the mutual ratios for ring plates of equal weight nmong themselves, but that we still disagree as to the ratio between entire plates . and ring plates of the same weight. The reason of this is.that in Mr. Randall’s discussion, whether he knows it or not, he hns not taken the conditions such that the total weight of his ring plates js the same as thnt of his entire plates; hut he has taken his ring plates sueh that their total weight is equal to tho weight of that portion of his cntiro plates which would be left if a circular portion whose’ radius is wero removed from their center. Tho total weight of his ring plates is, therefore, equal to ; of that of his ontire plates; and if his numhers for ring plntes he all multiplied by z tho corresponding plates will then all havo the samo weight, and our numerical results will agrce tbroughout. I had supposed that in my first article I had at . : Icast made my meaning so clear thnt a mathematician could not well misunderstand it; but I find that Mr. Randnll has done so, and there are in his . article some flagrant misrepresentations that I cannot allow to pass uunotieed. In speaking of the secondary eurve which I discussed, ho says: “Mr. Goodyear errs in premising . ’ that the wear to the plates takes place” (the itnlies are mine) ‘in the direction of the normal reekoned from the point of tangeney.’”’ I premised no such thing. By referring to page 162, he will see that I use tho words, “whero this wear would actually take place, if it could tnke place nt. all,” and the wholo tenor of my articlo went to show that such a wear could noé tako plaeo at oll, I discussed tho seeondary eurve, because I thoaght it involved in the proposition on page 103 of the Hand Book. If Ihave misunderstood tbo author upon this point, I regret it; hut I submit that tho man of straw was not entirely of my own ereating, and that the proposition referred to, if it means anything as there stated, may justly be taken to in. yolve the idea of this very wear which cannot take place, and, consequently, of ‘the “secondary, eurye which I discussed. " Again, on page 259, Mr. Randall says “ 1 Pa? is tho volume or size of the plate whieh Mr. Goodyear tells the pnblie has for its grinding effect 29° Pn* .” I did not tell them this; but I told them (and tho stntement is true) that, with circular plates of equal radius, aud having equal vertical pressures (P) nt the circumference, the grindiug effeet of plates,’ whose, hardness is uniform, is 22 Pa}, while if the hardness vary as y, it is 4 =n? Pa?. I distinetly stated, too, (and Mr. Randull has quoted the statement, with_an nceompnnying question that seems to me absurd, considering tho clenmess with. whieh«I stnted the standard of eomparison which I was using,) that in these two cases the otal pressure or weight (corresponding to what Mr, Randall here calls the “ volume”) was greater in the ono case than in the other. It is, in faet, twico as great, Also, on page 260, Mr. Randall says: “ Mr. Goodyear wodld show by his (fgures)-that a plane plate of uniform hardness will grind fifty per cent. more tlian a plate of the same size and, werght2 (the, italies are mine) “ and hardness‘at the eircamference, and which deeresges ‘ius hardness toward, athe eénter ‘in thé ratio of . the deereuse of ‘the radius.” ‘This is not what I said or dttempted to show. It would not be true. What I did say-will be apparent by referring to my article, and I was viele upon this points 4; On the same page lie says: “ z= tells us that the entire plate of any of the forms eonsidered will grind more than the corresponding ring plate of the proportions given, which I have proven by the authorities and demonstration to hen misstatement. He tells os that the plane plate of uniform hardness will grind more than Randall’s Patent Grindivg .Plates, which statement is doubly refuted.” Two more misrepresentations of the same kitid! I mado no such statements lor ‘plates olvequal weight. I made them only for. plates having equal pressures at ‘the eircumference, nnd for such plates hoth statements are trae, My comparisons throughout that articlo were for equal pressures at the circumference, and not for oqual weights of the plates, and I said o then. Mr. Randall's solntion of the problem of plane and conical grinsling plates, on page 260, if I understand it, tacitly admits the principle of the distribution of pressure which I have Fennnciated; for otherwise, his. proportion, “S$: .V:: EB: G” could uot hold trae for solid plates. He is approaching-tho truth.He has elready corrected various-errors in the Hand Book relating to ring formed plates; and when in his calculations and statements of comparative results he makes his ring plates really of the sume weights his eutire plates, he will he bearer right than he has been yet. If. it. would oof he saying too much, however, I would snggest that his demonstrations would be ug clear and no less elegant if he would not confound volume with pressure. The yolome of a solid of uniform density is, of course, proportional to its weight and to the total pressure: produced by that weight ; still, when dealing in fact. with pressure instead of . volume, it 1g just as easy to say “ pressure,” and in some cases it is less likely to lead to indistinct and, possibly, erroneots ideas, His misunderstandings and misrepresentations of my clearlyexpressed meaning evidently arose, witlf one exception, sdlely from his confusion . of the ideas of volume and pressure. ., ’ “It will be seen from what proceeds that I now admit that differently-formed plates of equal radii, hardness, and weight produce very different amounts of ‘grinding effect inn singlo revolution. ‘The tractory convid, however, ‘by no means gives n maximani of grinding effect nuder these conditions. ‘Ihe expression for thé grinding effect of the cone isa function of Phy the hight of the cone, and by increasing h, we can inerease without limit the theoretical . grinding effect, while still preserving the same . weight, eté., of the plate. But there is another point upon which I must say a few words, though Mr. Randall does not wish it “ logged into ” the question, ‘aud that is power or mechaniéal effect. ‘Lhe point which I had proposed to discuss was not simply the relative amount of grinding effect produéed by plates of dilferent forms. It was the broader qaestion of the relative advantages in the grinding effect to be derived from differentiy-lormed grinding surfaces, and into this question the element of power neces‘sarily enters. “It will he seen that formula (10), which gives ‘the mechanical effect or power eonsumed by the friction:in:a single revolution, and formula 11), which gives the grinding effect, pre identical, with the exception of. the. coelficient of frietion f, which is constant, and as there formulas nore geueral, the: law is also generul, whatever bo the form of the grinding surface, that the mechanical effect: or power expended is directly proporlional to the grinding effect produced. Mr. Randall, on pasé'260, after first objecting to any consideration: of the: element of power, says: “ No mention iu that work (the Hand Book) is made of power in connection with grinding effect. It requires no demonstration to prove that it takes more pozer to reduce ten tons of rock than it does to reduee one to the samo degree of fineness from the same condition at first. Any school-boy can tell us that, saying nothing of practiced miners and mechanics. What*was claimed, and is claimed, and is true, is that the tractory formed pan will do very mach more work than cither the flat or conical pan-of ithe same weight, andi I will uow add do the samo anlount of work at a less expenso of power.” The Hand Book, indeed, makes no mention of power, und, therefor 2; (in addition to its other errors,) the whole character of its dis