of Attractive Forces. 323 



account by merely substituting ku for a in the usual investigation 

 of the equations applicable to this case of motion, Ka being the 

 rate of propagation. Hence the solution of the proposed pro- 

 blem is given by the known equations, 



«VNap.logp+^ + ^'=0, 



~-=—Ka~=—KaN, 

 at dz 



it being supposed that V = where p = l. Hence putting 1 +0- 

 for p, and expanding to terms of the second order, 



KH^{a- ^'] - «aV + y = 0. 



Thus to the fii-st approximation Kaa=-N. Consequently, by sub- 

 stituting this value of V in the last term, the terms of the second 

 order disappear, and the relation between a- and V to the second 

 order of small quantities is still expressed by the same equation, 

 V = Kaa. 



This equation will accordingly be used in the subsequent in- 

 vestigation of the dynamical action of the assumed series of waves, 

 to small quantities of the second order. It may be remarked 

 that, as these waves are supposed to be such as result from the 

 mutual action of the parts of the fluid, there can be no perma- 

 nent motion of translation ; and the expressions for V and o- are 

 consequently periodic functions, having as many j)lus as minus 

 values. I have, in fact, found by an a priori investigation, given 

 in the Philosophical Magazine for May 1849, expressions for 

 these quantities of the following form : — 



m sin q{z — Kat + c) + Qm^cos 2q{z — Kat + c) . 



It may also be observed that, so far as the dynamical action we 

 propose to investigate depends on the square of the velocity, the 

 second term of this expression is insignificant, because it gives 

 rise, on squaring, to terms of a higher order than the second. 



Problem II. A smooth sphere fixed in position is submitted 

 to the action of a uniform stream : it is required to find the 

 pressure on its surface. 



In this case of motion, the velocity and condensation are con- 

 stantly the same at the same point of space, and are therefore 

 functions of coordinates only. Hence, taking the known equation, 



which is of perfectly general application so long as V is the total 

 velocity of the fluid, and ds is the increment of a line drawn 



Y2 



