the Resistance of the Air to an Oscillating Sphere. 233 



conditions of the proposed problem require that when R is 



d $ 

 substituted for r in -7-7, the equation 



d<f> . dc . 

 -r- cos 9 = -7- cos 9 

 d r dt 



should be true, whatever be 9 and ij. Hence it follows that 



4 is a function of r and t only, and that -r- = 0, -~- m 0. 



as at\ 



The above equation consequently becomes 



d 2 . r<p _ 2 d? . r \ 

 ~~df~ ~ a ' ~~dl3~> 



according with what precedes both in giving for $ a function 

 of r and t only, and in making sec 9 (u dx + v dy + *w d z) 

 an exact differential. Hence the value of the factor N has 



been correctly assumed. Since — = 0, and -— — 0, the 



a 9 at\ 



velocity is wholly in the direction of r, and V s= -~ cos 9. 

 But by integration, r <p =f{r — a t). Hence 



V = {/' <7 * ')-/('•-'*')} cos S. 



This is the value of V which I am contending for, and from it 

 the result I originally obtained respecting the resistance of 

 the air to an oscillating sphere necessarily follows. 



All the above reasoning, excepting the introduction of the 

 factor N, is according to the principles of the method adopted 

 by Poisson. The sole reason of the difference of result is, 

 that Poisson has argued as if u dx + v dy + w dz were an 

 exact differential in an instance, where, if my reasoning be 

 good, that quantity is not an exact differential unless it be 

 multiplied by a factor. 



The investigation I have now gone through is in perfect 

 agreement with the principles I advocated in my communica- 

 tion to the June Number of this Journal (S. 3. vol. xviii. p. 477), 

 and may serve to place in a clear light the position I have 

 there maintained, viz. that udx + vdy + id d z cannot in 

 general be an exact differential unless the variation of the co- 

 ordinates at a given instant be in the direction of a normal to 

 a surface of displacement. This limitation is equivalent to the 

 introduction of the factor N. 



Being aware that I am arguing for a principle,which, if true, 

 must have a very important bearing in the treatment of hydro- 

 dynamical problems, I feel it incumbent on me to support my 



