158 THE ROYAL SOCIETY OF CANADA 



^^'^" ~ ^ V^^^^ôT + a« • dudv J ~ "• 



On taking account of equation (12), it is readily seen that these 

 equations are satisfied identically. 



Equations (12) and (13) give rise to the following relations: 



gu = — i(t>uu-\-Q4>4>v), fv = — (0w + 6 (/) 0„ ) , 



_ 1 aMog (/) 1 a log (j) a- log 4) a log (/> aiogc^ „ 

 ^^ ~ 2 ^^3 ^ dv ' az'2 a?^ -ay ''^ ' 



, aMog(/) 1 a loge/) a'^ log ^ a log ^ ^ log </> . , ^ 



2(& -/«) = 

 . go, -fa'u 



-fvv ^ / <i>uuu-(i>vvv \ , 1 raiogcA a^ log ^ alog aMog^l 

 4, \ ct> /'^2<^[ az^ ■ a^az;'- dv ' dii^dvl' 



[ dHog(t) aMogc/) "! raiog</) aMog0_aiog^ a^iog ^l 

 du^ dv^ J I ^" ^^^' ^^ " ^^^ J' 

 ^ ^ r aiog0 a- log (^ _ a log^ a^iog^l 

 *" I a^ az^- a?; ' at»- J 



r / a log Y / a loge/) Y] 



The third integrability condition can therefore be reduced to the form 



, aiog</) aMog(/) _ aiog(/) a^ log ^ ^ 



aM ' az/ a^"^ dv ' du^ dv 



But this condition is satisfied identically on account of equation (12). 

 Thus the canonical system whose coefficients are given by equations 

 (11), (12) and (13) is completely integrable. 



Any system of equations obtained from a completely integrable 

 system by a transformation of the dependent variable will also be 

 completely integrable. In particular, the system of equations 



d'^y _^d\og4> _^ + 2c/)-^ = 0, 



(gy aw^ du 'du dv 



^-y _L 9 A ^y J- ^^Qg*^ ^y - 



dv du dv dv 



obtained from (6) by the substitution 



1 



y = 



X {îi, v) 



where \{u, v) is a solution of (6), is also completely integrable. We 

 shall call (6)' the Normal System. 



