518 



PROFESSOR KELLAND ON THE THEORY OF WAVES. 

 -^2 r+7 b, b, (e'^+»'«-- e~^"") cos V^ 6 



X {a(o,e"--e "" cos 6 + 2a., . e^"--e~"^''-'.co.s20+ . . .) ] 

 the symbol 2 applying to all values of r and s from 1 to oo . 



b, 2 a, 6j 



b, a, b 



3a3=T^*3, 4a, = -^6, &c. 

 0/ o. 



16. But since 

 Similarly 

 Hence, if we substitute these values in the above equation, it gives us 



+ =2 ir-s) brb, (£'• + »''- + £-'■ + »'•') sin »--*e 



-=2 {r + s) brb, (e'--'"- + e- '■-"-) sin r + ^ 



'] 



X {^-"■j (6,e"-'+e-"-siii0 + 6,.e^"- + e~'^"-'sin20+ ...)} 



:-r-^ + c^a{6,e"--e— «-sin0 + 2 6,.e^"--e-2''-sin20 + ...} 

 -^ 2 (>■ + «) 6, 6, (^g^«^_g-7T7..-) co8(r-«) 



+ " 2 »^ 6, 6, (/~---e-'^'"-') cos M^- ~j 



17. By multiplying the upper line of the first side of this equation by its 

 factor, we obtain 



(c-6„)a2/-6,(e"- + e-'""-) cosj-e 



_aa "l. (c-b„) •S.rb.b, (f- +e~''''') (e»«* +6-""-) cos r sin * : 

 b, 



the analogous term on the second side of the equation is 



-a2(c-6„)^ 2 (e'""^ _e-'-'"-)(e««^ -«-'«-) 6,6, »• cos*- sin >• 0; 

 6, 



if this result be brought to the first side of the equation, and combined with the 

 former, we obtain 



