380 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



Now clearly 



(hi 



'xj " dx t JZj 



The double sum in (75) thus becomes 



= 2 



dL, dL r 



r=2n dXj dx k 



,ni dM r 



dXj dx k 



r-0 



■^-\" dL, -^ dL, 7 



■Z «p dZ Aw. dx » 



p=0 



p=2« 



P=0 



dM, >^" dM r 



i> 



p=0 



'P dx a 



p=0 



dx, dX f> 



and since 

 this is 



dx: 



ITt = U J 



r-2n ( 



= 2 \ dMr 



r-0 ( 



Substituting this in (75) we have, after some reductions, 



r r=2n nr~\ r=2u / 



~df d2I >- 



dM r 

 -dt dL 







The equations 



diiQ 



du 



dlli, 



auij ««o ua o 



d.c 



dU du x dui du x . 



dJ, = Tt + M « ^ g + «i ^ + 



dC" 



tfefc 



rfifa 



(2ec 



UU u«o„ . «"2n . «"2n | 



<£*■<> 



<fo ^ f<1 die, 





, dun 



tfz. 



and 



(ft*,, </(/, rfWj . 



dxg "^ dx x 

 are thus replaced by a set of the form 



dL, 



dL r 





«& 



rfZ r 



+ 



+£-:=o 



rfi r dn 



<Of, . dM, . dM 



cfe 



dx 



dM r 



dM, _ dSl 



+ W 2« &~ — di r 



(76) 



(77) 



(78) 



(79) 



(78) remaining unchanged. In (79) r has, of course, all values from to 2n, and D, 

 is an arbitrary function of the X's, ilTs, and t. For U we have now 



