PROFESSOR KELLAND ON THE THEORY OF WAVES. 519 



(c-6„)a2>-6, (e'^-' + e-"') cos r 



+ a 2 (e"^" " + fi-*^" -") »■ '^r *» sin »^ 



-a2(e'^«* + e-'^"0 »■*.*« sin M^e (1) 



a. 



it being observed that ayc-6„=l by (14). 



18. The result of the factor unity is (omitting the part which has been al- 

 ready considered) 



-^2(r + s) brb, (e^' "- + e" "^^ «' ) sin ^+7 6 (2). 



The remainder of the upper side is 



__^ 2 (r-s) brb.b, (e^^+^'"+e-'^") (e«" + g-'"") sin V-^s 6. sin f 6 

 2 b, 



+ ^2(r + s) brb,b,(e'^"' = + e-'^'"-') (e*"-' + e-'") sia7+i 6 sin / : 

 and the corresponding terms of the lower side are 



a' a. 



^2(r + .)6,6,6,(e'- + ^«---e-'- + '«-') (e'«-- 



2 6. 



- e 



_^2(r-«) 6, 6, 6, («'-'«--«-'■-*''-) (e'"' -e-"") cos r + s0 coat 6; 

 which being brought over to the other side, the sum of the respective terms is 



2 6, L 



+ (r + s) [e +e + e + e ) sin (r + s) a smf a 



— {r-Vs){e —e +e —e ) cob (r — s) 6 con (6 



, / \ / r — s + tv.:: r—s—taz , — r — s + tuz — r — s — tac^ / , \ zl , zl 1 



+ (r — s){e —e +e —e ) cos{r + .i) 6cos fa 



which becomes, by multiplying out, and adding together, such quantities as will 

 make up cosines of sums or differences of the arcs which appear in the present 

 form, 



«" a 



r_ ;,(e'- + s + f«- + e-'" + '' + '"0' 



^ -'2 6,6»6, I _^(e'- + ' + ^«- + 6.-'^ + '' + "^ cos /■-«-/( 



g(e''^'*'" +e-''-''-^"")cosr-s + et 



+ r (/ + '-'"-" + 6-'' + '--'") cos /■-* + ^ 



+ s(e''^'-"'' + e-'' + '-''-')coar-s-te 



