518 PROFESSOR KELLAND ON THE THEORY OF WAVES. 





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

 16. But since b,^2a^ 2a,=^a,. 



Similarly 3 03= ^63, 4o,=^6, &c. 



0, b, 



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



+ 2 2 (r-s) brb, (e'- + »«' + e-'- + ««^)sm/-*6 

 -^ 2 (r + *) br bs {f^"^' + e-*^^"- ) Bin Ws 6 1 



-^ 2 (?■ + *) 6, 6, (e'^s«^_^-7T7«.>) cos(r-*) ^ 

 + ^ 2 »^^ 6^ 6, (e*^"*- e-^^«-' ) cos 7+^ ^ n 



a a 



l{6^e"^-e-«^cosO + 6,(e^"^-e-2«^cos20 + ...} 



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

 factor, we obtain 



(c-b,) a 2 /• 6, (e'-«^ + e-*""^) cos r 

 _a2^(c_6„) 2r 6, 6, (g*-"^ + g-*-*^ (eS«« + g-^"^) cos r ^ sin * ^ : 



the analogous term on the second side of the equation is 



-a2 (c-bo) ^ 2 (e*-"^ - e-*-"^) (f"' - e-««^) 6, 6, r cos * sin / ^ ; 



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

 former, we obtain 



