242 PROCEEDINGS OF THE AMERICAN ACADEMY. 





cJClo 0.2 



These equations give 



0— = — e*=, ;^ = «i eS n~ = ^' n~ = " ^ 



cyOi c'«2 cyfli c'a2 



and, inserting these values, equations (7) become 



0, 



-1, 



5«i 

 9a<2, 



e^s- — = e*2(a;2 - «i). 



5 02 



If in (8o) we insert tlie values of a;/ and a^g' derived from 



3C-^ — 37j ~r~ (z^j 



^2 = e^s X2 + Oi, 



they become equations between the independent quantities x^, X2, «i, 02, 

 bi, b.2 ; and must, therefore, be satisfied identical)}'. Hence, equations 

 (8 a) will still persist identically, in virtue of equations (1), if we assign 

 definite values to the ^'s. 



For this purpose let e^ = 1. We then have 



(9a)^r— = 0, -^ — = 1, -^ — = 1, ^ — =zx2—ai. 



dtti da^ dtti dao 



Therefore, if we define functions i of the x"s and functions 1// of the a's 

 as follows, 



$ii(x)=0, ^,„(x') = -l, ^,,(x') = -l, i,„(x')=-X2', 



xpn («) = — !, i/'i2 («) = «i. «A2i («) = 0, >/'22 ( '0 = — J 5 



we have 



