on the Resistance of the Air to an Oscillating Sphere. 67 

 Again, putting the letter Q for ^j^ — -^-, we have 



^ a R diso d.Q z*x dR x Rx 

 w = Q** + -; -^ = ^.— + j?- ^— 7s- 



ar r 



and m = Q z .r ; -3 — = -7— (- W #• 



~ dz dr r 



HenCe dx~d~z ~~ dxdt ~ ' dzdt~ dzdx' 



It is unnecessary to go through the other cases. 



This verification, it will be seen, entirely depends on the 

 fact that udx + vdy ■+■ 10 d 2 is an exact differential, and is 

 true only in the sense in which this analytical condition is 

 true. It is at this point that the method I have adopted diverges 

 from that of Poisson. From the foregoing mathematical rea- 

 soning, it appears that in Poisson's method that condition is 

 not satisfied independently of the particular circumstances of 

 the motion, and of a particular function, cos 0, introduced by 

 considering the nature of the arbitrary disturbance. On the 

 contrary, I argue that as it must be satisfied prior to the con- 

 sideration of any instance of motion, it must be true inde- 

 pendently of all that is arbitrary ; and I have shown in the 

 Number of the Philosophical Magazine and Journal for June, 

 that the quantity in question is in this manner an exact dif- 

 ferential only when the coordinates are restricted to vary from 

 one point to another in the line of motion. The function <p 

 may thus contain implicitly as a factor another function ex- 

 pressing the variation of velocity at a given instant in passing 

 from one point to another in directions perpendicular to the 

 motion, but is not differentiated with respect to the variables of 

 this factor. Accordingly the equation for finding the value 

 of <p applicable to the motion caused by a vibrating sphere is 



dt 2 dr* ' " r 



the factor, cos 0, depending on the mode of disturbance, being 

 included in the arbitrary function. The verification of the 

 six equations by the above expression for $ proceeds in the 

 same manner as before, but by simpler operations. As <f> does 

 not contain explicitly the angular coordinates, the velocity 

 perpendicular to the radius vector is nothing, and the whole 



F2 



