345 



therefore we suppose that the initial data £ , and £' 



g> J>> g, h, 



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

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



Z, — x\ a\ _ cosft' — S..,V.# X (14) 



g, «, 1 h,\ \. 1 (*)i % g,ty ^ ' 



% 7 n— ~ s \ x \ A \ , sin C s r — 2 "«\# .Y (14)' 



g, II, 1 1 k,\ \ 1 (;)1 % g,lj' * ' 



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

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

 the general expression (9) to the particular form (4) v ; 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 



, f 2C C °smCA;2 " u x ) . \ 



i f I - - J V m ' K>*' die J , (15) 



which is — 1, or — i, or = 0, according as the sum 



2,. "u.x . is < 0, or z=. 0, or> 0. It is found that, in this 



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 : 



% , , = J x\ a\ . cos(Y.. 4- ts\ — S, "u.x .\ ! 



g,h,t 2 1 M \ 1 ' 1 (z)l l g,l) j 



in which 



l — p F cos lex — q k sin 7i\r, 



p # sin 7ur -f- Q £ v cos &£, 



^ (17) 



