343 



s l A, = S, "h, .a., (6)' 



h (i)l fl, » » v ' 



in which, by (7)', 



It is important to observe, that by the form of these equa- 

 tions (6)', (which occur in many researches,) we have the 

 relation 



2 "a. a. =0, (5)' 



(ft) 1 h, q h, r * v ' 



if <? be different from }• ; and that, by (5) and (5)', we have 

 also the relations 



\»\r\r = °' ( 8 )' 



In the particular integral (4), we may consider u., . . .u 



1 n 



as arbiti'ary parameters, of which x and c are real and ar- 

 bitrary, while s 2 and a. are real and determined functions ; 



■« r h, r ' 



and hence, by summations relatively to the index r, and 

 integrations relatively to the parameters u., employing also 



the relations (5) (5)' (8) (8)', and Fourier's theorem ex- 

 tended to several variables, we deduce this general integral, 

 applying to all arbitrary real values of the initial data : 



%^M u (oT_j^(\ t cos +\ t sin ) *«>«**'' (9) 



in which 



\ t = \)l \ r ( V r C0S K + Y 'r C shl tS r)> 1 



-1 [ ( U > 



\ t ~ \)l \ r ( Z r C0S tS r + Z 'r S r sin K) '> j 



