SLOCUM. — FINITE CONTINUOUS GROUPS. 245 



9 a. 



9fij 



_ 1 / eM2- 1 \ 



is not identically zero. 



By means of equations (12) we can introduce in equations (1) the 

 new parameters fii, fi^ instead of Ui, a^. Solving (12) for /^i, jMo) ^e have 



(«2-«2)(«i- -^je"^-"^ 

 H = i-e^.-cH = ^^ (''' «)' 



fxo = 02 — a2 =. Mg (a, a). 



If we introduce these new parameters fxi, /^.o in equations (1), the ar"s 

 become definite functions of Xi, x.^, fii, /j-z, «i5 «2 5 3,nd, since the fi's contain 

 t, the a;"s will also be functions of the auxiliary variable t. Multiplying 

 equations (10) by Ai, X^, and adding, we get 



4 ^ e / ' 'A % 9x{.daf, 



2,A,4.(x, .,) = >. 5^^ (^ = h2), 



or 



dx' 



—jj' ^^ '^1 ^11 (•'^l J ^2 ) + ^^2 ^21 (3^1 J a^2 ) = — ■^2> 



(14) 



-^ — ^1 ^12 (a:*/, x.{) + A2 ^22 (a?/, xl) = — Ai — A2 a^a'^ 



which define the a;"s as functions of t. Since «!, Oo reduce to Ox, 02 for 

 < = ^, we see that x{^ x^'-, for < = 7, take the initial values 



Xi = a?! + «„ =/i (a-, o), 

 (15) _ _ _ _ 



a^o' = e^-'-Xo + «i ==^2 (^1 «)• 



Further, the integrals of equations (14) contain the A's only in the 

 combination /^i, /^o ; and, therefore, can be expressed in the form 



Xx =: -Tl {X-x , X<2, , fXi, ^2) = a*! — /X2, 



(16) 



, XT / — r ~ r - A''! f 111 '^'' 



a:2 = i'o (a:^i , a^2 » /^i, /^2) = — ( — 1 + —• 



Inserting therein the values 



^/ =/• (^» «)? '< =/• (^' «)j ^k = <^4 (ft, «) 



(i = l, 2; ^-=1, 2), 



