345 



therefore we suppose that the initial data ^ , „ and ^' 



' ' g,h,0 ^ g, h, 



are all such as to agree with this particular solution, that is, 

 if we have, for all values of ^ and h, 



iM=^\^M<=°HS-S(,V'>„,). (1*) 



we see, a priori^ that the multiple integrations ought to 

 admit of heing all effected in finite terms, so as to reduce 

 the general expression (9) to the particular form (4)^; an 

 expectation which the calculation, accordingly, a posteriori^ 

 proves to be correct. An analogous but less simple re- 

 duction takes place, when we suppose that the initial 

 equations (14) and (14)' hold good, after their second mem- 

 bers have been multiplied by a discontinuous factor such as 



, /, 2 f ""sin f A; 2 " w .r ) , \ 



ill \ ^ ^'^1 '• s^'^ dJc ) , (15) 



V TT^O k J 



which is z= 1, or =z J-, or •=. 0, according as the sum 



S,.,%\a7 . is < 0, or z: 0, or> 0. It is found that, in this 



{i)\ t g,i ^ ' 



case, the 2n successive integrations (required for the general 

 solution) can in part be completely effected, and in the 

 remaining part be reduced to the calculation of a simple 

 definite integral ; in such a manner that the expression (9) 

 now reduces itself rigorously to the following : 



^g,h,t ^ 1 h,l VI' 1 (i)l t g,i) I 



\ C^ dk I 



in which 



L^ — p Ic" cos kx — Q k sin kx, 



M^zz p^k sin A\r + q^ A-^ cos ^o;, 



V (IG) 



(17) 



