343 



in which, by (7)', 



H. , = H, .. (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)' 



(A)l h,q h,r ' ^ ^ 



if q be different from r ; and that, by (5) and (5)', we have 

 also the relations 



(r)l h,r ' ^ -^ 



2,,%, A. =0. (8)' 



(r)l h,r t,r ^ ' 



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

 as arbitrary parameters, of which x and e are real and ar- 

 bitrary, while s^ and a ^ are real and determined functions ; 



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 : 



in which 



nco ^00 pco poo 



"(.■)ij '^"1=) '^"0 *2---3 ''«„; (10) 



"— so " — 00 •'— 00 •'^■«» 00 



E, .=: S, .^A^ /Y COS ts 4" y' s~ sin is \, -, 



h,t (r)i h,r{r r' r r **/ 



F, =. S, x%^ /z COS ^5 + z' s sin ts \: 



(r)l ft>rlr r' rr ^) J 



