36 Mr. R. Moon on the Solution of Linear Partial 



dxdy dxdy dx dy^ 



(. du du dA du dA du . d 2 w \ . ]( . 

 K UxTy+tohj+a^Tx+ K JxTyF W 



/ . du du dA Y du dk x du . ^% c? 2 A \ , > 

 + V A2 ^^ + ^^ + "^ S +Al dxdy + dxdyJVW 



(. </w e?w dA 2 du dA 2 du . c? 2 w ^AA . } , . 

 ^Tx&j + TxTy + ~a^Tx+^Jxdy* dxdy) 9 W 

 + &c; 



and we shall have an expression for -7-5 identical with that for 



-7-5 when in the latter we put y for #. 



Substituting these values in (1) and equating to zero the co- 

 efficients of <j)"(u), <£'(w)j <j>(u), <£ -1 (w), &c. ; we get 



_ ^d^A c'd 2 A . m^ 2 A r>dA „dA ttT , r /ON 



0=R ^ +S ^ +T iv +P ^ +Q ¥ +UA+V '' ^ 



0=R^r+S^^+T§ 2 , . . (4) 





1 f R — I S d ^ u 1 T^" I P rfM 1 O du \.\ 

 + K d** + dxdy + L dy* + F S + U ^j A ' 



\ dx ay J ax \ ax dyj dy 



H K » +s Sr +T $ +p £+ a t> 



= f 2 R^ + S^^ + fs^+2T^^ 

 V dx dy) dx \ dx dy) dy 



&c. &c. 



