SLOCUM. — FLNITE CONTINUOUS GROUPS. 107 



If this transformation is denoted by T^^^ then from the symbolic equation 

 Tf,Ta^^ T^ we obtain the five relations 



(28) e-3 »^i _ ,- <-3 * ~i == I (e'3 ^^i + ,- 's * ^^ (, s v " i _ ,- "3 v^) 



(29) c, = a,-]-b,, 



(30) ^ (."3 * - 1 + e- '■3 *'- 1 - 2 ) - ^^^ (/3 *'- 1 _ .- '-3 « - 1) 

 2 C3 2 C3 



= # («''^'~' + e- '^'^ - 2) - ^L^;^ (,'3^-i _ e-*3v-i>> 



^3 ■^ •'a 



X ( :^ (e"3 *'-! + .- "3 ^'^1 - 2) - ^i^' (/3 ^ =^ _ ,- "3 ^'^) I 

 I2a3^ ^ 2a3 ^ ^j 



+ ^(e''-''-'-e-^^^-') 



2 



X 



{-'-^("^■■'-'--"*^)-2^/'"'''-' + »-'''"'-^)}=X. 



(31) ^(/.'-> +,-'.'-l_2) + ?l^^(<!'.*~l-.--.'-') 

 2 C3 2 C3 



= ^(^'^''' + «-"^'-' - 2) + ^^^ (,^3*'-i_ e-'^3V-i) 

 2 03 z 03 



V 2 ^3 2 (3(3 J 



+ 2^^ — e 3 ) 



Denoting the right-hand members of e(iiiations (oO) and (31) by x '^^^^ 4' 

 respectively, and solving for Ci, c^, c^, we have 



