Partial Differential Equation of the Second Order. 225 

 Hence (8 1 ) becomes, putting m l -7- for -7-, 



o=/'.W-%/'.W +/.•/'» +«L+v*+* 



du 



+ m-m 2 .^./' a (U_ 1 ).U 



+ {^ -AW +«.«+^+^^} ./. 



Now /«> /'»> /'»; /.(*) are all free from U, while / con- 

 tains U. 



If / contains U otherwise than in the simple power, the 

 coefficient of / in the last equation must vanish, since no other 

 term contains U otherwise than in the simple power. 



Moreover, if the coefficient of / vanishes, the coefficient of U 

 must also vanish; so that we must have either / / a (U_ 1 ) = 0, 

 which reduces this case to case (I.) already discussed, or we 



must have m^m^ or -y-=0, a particular case which may be 



passed over for the present. 



We may confine our attention, therefore, to the case where / 

 is linear with respect to U, i. e. where 



f{xyz\J) =f p (ayz) U +/ y [xyz] , 



where /„, / do not contain U, but may involve lower derivatives 

 of <£. Substituting this value in the foregoing equation, and 

 equating to zero, the coefficient of U in the result, we shall have 

 simultaneously 



= m,f a (y) -/» -f a .f a (z)-*f-yz-S 



y 



°=J/'«( U -.) +(/'» -Mr 



}■ 



(13) 



where 



1 f , o . d }7l i dm x \ 



Moreover, substituting in (8 U )* the following values of f x ,f y3 



* It is to be observed that although terms in/^,/, involving U_i, 

 would give rise to terms in f x ,f ' involving U, yet those would disappear 

 when we substitute ioi'f x ,f in (8 M ). Hence in effecting such substi- 

 tution we need not advert to the fact of whether f^ f do or do not 

 involve U-i. 



Phil. Mag. S. 4. Vol. 35. No. 236. March 1868. Q 



