SLOCUM. — FINITE CONTINUOUS GROUPS. 245 



9a k 



_ 1 / >! — 1 \ 



is not identically zero. 



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

 new parameters ju, l5 /x 2 instead of a u a 2 . Solving (12) for {jl x , |U 2 , we have 



(cio — a.,) [ «i — - — ) e"2 — a 2 



V ^ 1 e a 2 -a 2 J _ 



>*i = T^ . ««,-«, - = Mt(a,a) f 



(13) _ _ x 



Hz = a 2 — a 2 — M 2 (a, a). 



If we introduce these new parameters [x x , /x 2 in equations (1), the x"s 

 become definite functions of x u x 2 , ^ l5 fi 2 , a x , a 2 ', and, since the //,'s contain 

 £, the x"s will also be functions of the auxiliary variable t. Multiplying 

 equations (10) by A l5 A 2 , and adding, we get 



2 2 3x1 da, 



jw^^ki (i=1,2) ' 



"1 fell ("''l J ■'-S / T Ao f 2 l (Xj^ , Xo ) Ao, 



or 



dx\ 



dt 



(u) **,< 



—t^ = A x £ i2 (x/, x 2 ') + A 2 £ 2a (x/, x 2 ') — — A x — A 2 x 2 ', 



which define the x"s as functions of t. Since a 1} a 2 reduce to a 1} a 2 for 

 * = t, we see that x/, x/, for £ = 7, take the initial values 



Xi = x x + a 2 =fi (x, a), 



05) 



x 2 ' = c a 2X 2 + a x z=f 2 (x, a). 



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

 combination [i u /x 2 ; and, therefore, can be expressed in the form 



X X = F x (x/, X 2 , [X X , [X 2 ) = x/ — fl 2 , 



(16) 



- «! / 1 \ X 2 ' 



x./ = ^ (x/, x/, H , fl 2 ) = £{— - 1 J + -• 



Inserting therein the values 



as/ =/< (x, a), x/ = /, (x, a), o A . = <J> t (jx, a) 

 (i = l,2; i=l, 2), 



