226 Mr. R. Moon on the Integration of the General Linear 



fa»> viz - 



ay 



dV du 



^ •*■• i +/ ^ w +WW+/J • u 



+/ y (#)+/ y (*)to/+/ji 



rfw ' dy 



fffn-x'X+W+M-f'V 



and equating to zero the coefficient of TJ in the result, we shall 

 have simultaneously 



0=mj y {y) -f y (x) -/„ ./» +/'» +/ y ./'„(-") +/ y • ^i 

 0-*/, (y) -A(y) -/. ./,W +/„-AM + J/.(U_0 +/,.^. 



Eliminating /* (tT_i) by means of (13"), the two last equations 

 may be written 



o=™,/' v (2/) -AW -/. ./,« +/.W +^./ v +/.W. ~" 



7 > (W) 



0=m l /,(y)-/ p ( a? ) -/../,W+(f» + 5')./ r 



From (13) and (1-1) we have to determine f a , /., /. 



From what has immediately preceded, it will be evident that 

 the forms of (13) and (14) will be precisely the same, whether 

 fp f contain <£ or any of its derivatives or not. Hence we may 

 assume that/*, f contain neither cf> nor its derivatives. 



But if/„ do not contain U_i, since f a contains U_!, andm T , fi 

 are definite functions of x and y, (14 u ) shows that we must have 

 fJz) = 0. Hence we have for the determination of/., 



o=%A(y) -AM +(/*+|£)/,,. (14*) 



Moreover, integrating (13 u ), the auxiliary equations for which 

 are 0=d.f a -p,dz, dx=0, 



0=1^-/^.11^, dy=0 } 

 whence we have 



<W.- W o f ,-J»-/.u_ 1< 



we get /«=/« +/i(ayo>)j (15) 



