242 PBOCEEDINGS OF THE AMERICAN ACADEMY. 



k 9 hi , c b n 



C "i J "i 



and, inserting these values, equations (7) become 



9*1 A 



(8 a) 



e"= ^ = e\ (x/ - a,), 

 c a 2 



If in (8«) we insert the values of x/ and x 2 ' derived from 



*l' = *1 + «S> 



x 2 = e"jx 2 + "i, 



they become equations between the independent quantities ./-,. r. : , a u a*, 

 b u b,; and must, therefore, be satisfied identically. Hence, equations 

 (Si/) will still persist identically, in virtue of equations (1), if we assign 

 definite values to the b's. 



For this purpose let e 6 « = 1. We then have 



, n N 9r x ' „ 5x/ , 3x a y , 3a / 



/ 9<i\ c c, c ", c ". 



Therefore, if we define functions £ of the x"s and functions <// of the a's 

 as follows, 



in V) = 0, & 8 (*') = - 1, &, (*') = - 1, &» (V) = - *,', 

 we have 



