230 Mr. B,. Moon on the Integration of the General Linear 



whence it is clear that we may put 



A=A 1+ 4.U_ 2 , 



where A l} f p are functions of ocij, which do not contain U_ 2 or 

 higher derivatives of <p ; 



.../.=Ai + S+? w . z-f H . v_+f h v_ 2 , 



Substituting these values in (13 1 ), also putting O + D for 

 / , and equating to zero the coefficients of z, U_ 1 and U_ 2 in the 

 result, which last we may do on the assumption that A l does not 

 contain U_ 3 , we shall get the four following equations, viz : — 



o=^|(^+^ l )-|(^+4)-.(«+f'+^ J )>+ 8 / ft )-y 



Also, treating in the same manner (14 1 ), we get 



d~D d~D ( t j, . dm x \^. ri . '. dA, 

 °=^%-&-(' t + e 4+-^)D-CA 1 + ^ 



dC dC dm d , t r , 



0=%-/ ( ,/( 2 /)+/ fe J-> 

 0=%-/^). 



Of these ei^/ equations, two may be disposed of by putting 

 A = D = 0, having also 8 = 0. 



By properly combining the remaining six we shall get a par- 

 tial differential equation of the first order by which f Pl may be 

 determined, and so obtain values for the remaining dispo- 

 sable quantities/^ and C, there being left a residuum of three 

 equations of condition to be satisfied by the coefficients of the 

 given equation. 



