II. ELECTRICAL RHEOMETRY. 9 



In order to find the components of this force parallel to the co-ordinate axes, the 

 rule already given is to be followed. Let us take the equation of the surface of 

 the oblique cone with a circular base, which in our case will be^ 

 H= [pz — xn)'^ + {inz — yny — R- (2 — iiY, 

 diflferentiating this equation successively, with respect to x, y, and z, and making 

 2 = after the differentiations, because all the points through which the normal 

 lines pass are in the plane xy, we shall have the following values of the partial 

 differential co-efficients — 



-^=2r^x^2irR sin. o, 

 -^ = 2 n^ y = 2 ?i^ i2 COS. (J, 



-^ z=2 n {R^ — px — my) = 2 nR {R — j5 sin. a — m cos. cj), 



^'" = 4 n^ R^ \ji^ + (R — p sin. o — m cos. o)^]. 



Hence the values of the cosines of the angles made by the normal line with the 



co-ordinate axes, will be 



n sin. 

 Cos. (Nx) = / ., , r, ■ =.,' 



"^ n + (it — p sm. 6) — m cos. uf 



„ , ^^ n COS. 



Cos. (Ny) = . , . - > 



'<^n + (it — p sm. 6) — m cos. o)- 



R — p sin. 6) — m cos. cj 

 Cos. {M) = . ,^ ,p ■ =f^ 



'^n + (it — p sm. a — m cos. o) 



and the three components of ^ will be 



rk n R sin. ad a 



{R- + P — 2 Rm cos. o — 2 Rp sin. 0) f 



1„ Choi R cos. ad cd 



(9) \^ J {R- + P — 2R m cos.a — 2 Rp sin. a) I 



„ rJc R {R — m cos. a — p sin. a) da 



<^ [R^ + P — 2 Rm cos. o ■ — 2 Rp sin. o)| 

 where the integrals are to be taken between the limits o = 0, o = 2 7t. 



The formulte (5) are general, and include as a particular case those given by 

 Savary in page 23 of the memoir already quoted, as will appear by making p = 0, 

 which gives the case of Savary. Indeed, in this case they may be written 

 rkn R sin. ada „_ rJcnRcos.ada 



„ rJcR (R — m cos. a) da 



which are the formulae of Savary. The first integrated becomes = 0, as is evident 

 from the symmetry of the figure in such a case. 



' See Moigno, Traits de Calc. Diff., I. p. 423, and others. — Davy's Elem. of Anal. Geom., p. 313. 

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