109-110] 



CONFOCAL QUADRICS. 



159 



^ 2 =4 



.(6)*. 



, 2 _ . 



0, 



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

 written down from symmetry. 



Substituting in Art. 108 (4), we find 



4 (i. - X) 



+ (X - /t) [(a 2 

 I 



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



110. The particular solutions of the transformed equation 

 V 2 &amp;lt;/&amp;gt; = which first present themselves are those in which &amp;lt;/&amp;gt; is a 

 function of one (only) of the variables X, ^, v. Thus &amp;lt; may be a 

 function of X alone, provided 



(a 2 + X)* (6 2 4- X)* (c 2 + X)* d&amp;lt;/dX = const., 



whence 

 if 



.................. (2), 



being chosen so as 



the additive constant which attaches to 

 to make vanish for X = oo . 



In this solution, which corresponds to &amp;lt;f&amp;gt; = 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 



* It will be noticed that h lt h 2 , h 3 are double the perpendiculars from the origin 

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



t Cf. Lame, &quot; Sur les surfaces isothermes dans les corps solides homogenes en 

 6quilibre de temperature,&quot; Liouville, t. ii., (1837). 



