158 



PROBLEMS IN THREE DIMENSIONS. 



[CHAP. V 



109. In the applications to which we now proceed the triple 

 orthogonal system consists of the confocal quadrics 



a 2 -i- 8 ^ 6 2 4- c 2 + 6 



-1 = 



(1), 



whose properties are explained in books on Solid Geometry. 

 Through any given point (x, y, z) there pass three surfaces of the 

 system, corresponding to the three roots of (1), considered as a cubic 

 in 0. If (as we shall for the most part suppose) a &amp;gt; b &amp;gt; c, one of 

 these roots (X, say) will lie between oo and c 2 , another (/JL) be 

 tween c 2 and 6 2 , and the third (v) between 6 2 and a 2 . The 

 surfaces X, /JL, v are therefore ellipsoids, hyperboloids of one sheet, 

 and hyperboloids of two sheets, respectively. 



It follows immediately from this definition of X, p, v, that 



a 2 + 6 6 2 + c 2 + 



identically, for all values of 6. Hence multiplying by a 2 +6, and 

 afterwards putting # = a 2 , we obtain the first of the following 

 equations : 



(a 2 + X)(a 2 + ^)(a 2 + z.) , 



(6 2 -c 2 )(6 2 -a 2 ) 



(c 2 + X) (c 2 + ft) (c 2 + v) 

 (c 2 -a 2 )(c 2 -6 2 ) 



.(3). 



These give 



dx 



d\ * a 2 + X d\ 2 6 2 + X 

 and thence, in the notation of Art. 108 (2), 



(4), 



X)&quot; (c 2 + X) 2 



(5). 



If we differentiate (2) with respect to 6 and afterwards put 6 = X, 

 we deduce the first of the following three relations : 



