PROFESSOR KELLAND ON THE THEORY OF WAVES. 519 



(c-6„)a2y6, (e''"* + e-''«0 cos / 



+ a2(/ + *"^ + e-'- + ««0^6r^.sin/-*0 



-a^{e''~"'' + e-'-"'')rKb,smr + s6 (1) 



it being observed that a-^c-b„=l by (14). 



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

 ready considered) 



J l(r-s) br b, (e'-+'''^ + e-'' + "'^) sinr-s e 

 -^2(r + s) brb, {^' "^ + ^-^^« «' ) sin 7+1 6 (2). 



The remainder of the upper side is 



2 b 

 a? a. 



'-l,{r-s)brb,bt(/-^"''+e-'- + "'') (e<«^+e-*"*)sinr-*0.sin/0 



+ ^2(r + *)6,6,6,(e'-^«" + e-'--^«") (e*«~' + ^-*«^) sin r + * 6 siiW^: 



and the corresponding terms of the lower side are 



a? a 



2 b 



'-l.{r + s)b,b,b,(f + "''-e~'' + "'') (e^""-' - e-*"^) cos r-s e cos 1 6 



i2(r-*)6,6,6,(e'-^«^-e-'"-*«^) (e*"' -e-*''') cos r + s 6 cos 1 6 ■ 



2 b, 

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



or a 

 26 



r — s + ta.z , r — s — ta.z , — r — s-vtuz — r — s- — ta.z 



+ {r + s) {e +e +e +e ) sin (r + *•) 6^ sin ^ d 



r + s+taz r + s—ta.s — r+S + t«z —r + s — taz^ 



— {r + s)[e ^ —e +e ~e ) cos(r — s) 6 cos tS 



, / \ / r—s + taz r—s—taz , — r — s + taz — r—s—ta.Z\ , , /i /, | 



+ {r — s)(e —e +e —e ) G0s{r + s) 6 cos fv 



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

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

 form, 



_^^^r + s + t.z _^-r + s + t.z^^^^^_^^^Q 



+ y (/+'-* " ^> 6-'' + ''-*"^) COS y -* + ? a 



+ «(/ + *-*«^ + e-'- + *-'''Ocos/-*-;0 



