on the Resistance of the Air to an Oscillating Sphere. 6.5 



The differential coefficients are all partial : and in any case 

 of fluid motion to which these equations apply, the following 

 equations must be verified : — 



m *!_--.*£- a) 



} u > dt dx " dx dt •••* v :{ 



d*P <F-P . 



( 2t > dt dy ~ dy dt "" f ' 



d*P d?P . . 



dzdy " dy dz "" ^ 3,) dtdz dzdt * f t 



The three equations (l.)> (3.), and (6.), are the same as 

 Mr. Airy's. Now it appears from equations (a.), (b.)> (c), 

 that the equations (1.), (2.), (3.) are at once verified, if 

 du _ dv du _ dw t dv _ dw 

 dy ~ dx* dz " dx* dz dy ' 



that is, if udx + vdy + wdz be an exact differential of a 



function of x, y, z t which may also contain t. Let $ be this 



function : so that 



di> d& d<p 



u — -j^- ; v = -j— ; TO = —r- • 

 dx dy dz 



Then, as Poisson has shown, [Traite de Mecanique, torn. ii. 

 p. 687, 2 e edition) it follows, to the same degree of approxi- 

 mation, that P + k. -~- = 0. Hence 



(I v 



d*P d?<p d*u rf 2 P 



dxdt 'dxdt 2 ' dt* " dt dx' 



by equation (a.). 



Thus equation (4-.) is verified: and so for equations (5.) 

 and (6.). It has been shown, therefore, that for the verifica- 

 tion of the six equations, it is necessary and it is sufficient that 

 udx + vdy + to dz be an exact differential of a function of 

 x, y, z, and Y, with respect to the three first of these variables. 

 On the same condition the equation (d.) becomes (Jc a" being 



VL$£i^$ri*& (a ° 



Hence if this equation be made applicable to a proposed in- 

 stance of motion, and a value of $ be obtained from it by in- 

 tegration, it follows from the preceding general reasoning, 

 that the same value of <p will satisfy the six equations above, 

 the only condition necessary for their verification being in- 

 volved in that equation. This will more clearly appear by 

 taking a particular instance ; and as Mr. Airy's argument 

 is not more opposed to my solution of the problem of resist- 

 Phil. Mag. S. 3. Vol. 19. No. 121. July 1841. F 



