z is an ex- 



234? On the Resistance of the Air to an Oscillating Sphere. 



views by an accumulation of evidence, and will therefore add 

 another argument. The equations (a), (b), (c), quoted above, 

 are accurately true if the complete differential coefficients 



K^Tf)' \d~tr \'JT) k e substituted for the partial differ- 

 ential coefficients -5-', -=-, -5-r, and become by this substi- 

 d t d t d t 



tution 



. dV, ,dV J rfP, . 

 Hence at the same time that -3 — dx -\ — j— dy + -3 — dz is an 



dx dy u dz 



exact differential, (^~)dx + (£?) ety + (^)rf 



act differential. Let therefore this latter quantity be the differ- 

 ential with respect to x, y, z, of a function \p of .r, j/, z and /. 

 Then 



(du\^d<\> t /dv\_d^ /diio\ _di\> 

 \di) ~~ d# ; \J~t) "~ 5^ ; V7/7/ ~* d~z 

 Hence, by integration, 



J dx J dy s J dz 



Consequently, differentiating independently of the time, 



du_ rd*ty pd*j> ' dv 



dy Jdydx ~J dx dy dx' 



„ du dw , dv dw 



00 -5— = -s — , and -7— = -= — . 



dz d x dz d y 



,rfP, ■ rfP., dP J 

 Hence tidx+vdy + wdz and —, — dx+ -: — dy-{- -= — dz 

 J dx • dy dz 



are also exact differentials at the same time. Now any one 

 arguing according to the views maintained by Mr. Airy would 

 say that the latter quantity is unconditionally an exact differ- 

 ential ; and how the conclusion could be avoided that u dx 

 + vdy -f wdz is without limitation an exact differential in 

 every instance of fluid motion whatever, I am unable to see. 

 But this conclusion is certainly untrue. Poisson has written 

 a long memoir on the propagation of motion in elastic fluids, 

 on the supposition that that condition is not fulfilled. (See 

 Memoirs of the Paris Academy, torn. x. 1831.) The expla- 

 nation of this apparent contradiction I believe to be as follows: — 

 When the complete differential {d P) is substituted for the sum 



