109-110] CONFOCAL QUADRICS. 159 



A,' = 4 ( ^ 



.(6)*. 



The remaining relations of the sets (3) and (6) have been 

 written down from symmetry. 



Substituting in Art. 108 (4), we find 



+ (v - X) (" 



+ (X - /t) [(a 2 + j/)* (&" + v )l (c 2 + v) 

 I 



............... (7)t- 



110. The particular solutions of the transformed equation 

 V 2 < = which first present themselves are those in which <f> is a 

 function of one (only) of the variables X, p, v. Thus (j> may be a 

 function of X alone, provided 



(a 2 + X)* (6 2 + X)* (c 2 + X)* d<j)/d\, = const., 



whence ^ ^1 ~\~ (1)> 



if A = {(a 2 + X)(6 2 + X)(c 2 + X))* (2), 



the additive constant which attaches to </> being chosen so as 

 to make <f> vanish for X = oo . 



In this solution, which corresponds to </> = A/r in spherical 

 harmonics, the equipotential surfaces are the confocal ellipsoids, 

 and the motion in the space external to any one of these (say that 

 for which X = 0) is that due to a certain arrangement of simple 

 sources over it. The velocity at any point is given by the formula 



d<t> , d<l> _ r A! ( , 



T = H\ ~jT = O -r- ( O ). 



dn d\ A 



* It will be noticed that 7^, h 2 , h s are double the perpendiculars from the origin 

 on the tangent planes to the three quadrics X, /x, v. 



t Of. Lam6, " Sur les surfaces isothermes dans les corps solides homogenes en 

 Squilibre de temperature," Liouville, t. ii., (1837). 



