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



nish. Hence we have simultaneously 



0-//(.)-«v/l'(y) + ( M +aj/i+8, . . (11), 

 ~ du, dub , . 



the last of which is an equation of condition which must be sa- 

 tisfied by the coefficients of the given equation in order that the 

 latter may admit of an integral of the kind treated of in this 

 paper, under the circumstances we are now considering, i. e. 

 when/ ft does not involve U_i. 



A reference to my former paper will show, it being observed 

 that the m there used is the reciprocal of that above employed, 

 that the above equation of condition is identical with that which 

 must be satisfied in order that the given equation may be capa- 

 ble of being derived from a single partial differential equation 

 of the first order. Also 



in other words, O^p-mtf-ftz-fv (12) 



where f b is determined by (11). My former paper shows that 

 (12) is the single partial differential equation of the first order 

 from which, the condition being satisfied, the given equation 

 is derivable. 



II. But if f a contains ^ m ~ l ) or U_!, the process must be 

 modified. In this case, if 



f a {xyz) =/«(a?y*U_i)*, 



weshallhave fJ(^/J^H/XV^.V i |> 



//(y)=/« , ( 2 /)+/AU-i).u/J 



. du ^ /«'{*)=/*(*), 



since — =0"j\ 

 dz 



* Note throughout that when a symbol is used as a subscribed index, and 

 also as a symbol of quantity, no connexion exists between the two modes 

 of user. 



•■ d(u) d(u) n . 



t This appears thus: -^ ■m 1 "^r=0 is equivalent to 



dz dz 



the integral of which, when m x involves x and y only, will always be of the 



form s=funct.{#,y, <£(«)}, 



where o>=a const, is the integral of 0=dy-\-midis; whence it appears that 



du 

 the assumption that ^ does not vanish leads us to an integral relation be- 

 tween sn, y, % different from that which we are now supposing to exist. 



