526 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



25. Also, u = b sin ^r--(ci—x) 



J. . ' 27r , 

 A 



du ,27r 27r, 



ds 2nr 27r, 



-r— = — « • "s— cos . -.r— (c if— a;), 



-^ =-^ be cos 6 J- b^ sin ^ cos 6. 



Consequently, -^ be. cos 6 — ^ b^ sinOcosd = — a^cosd 



A. A X 



__ 277-^2 « COS 6sm^d + sin ^ cog 6 {h + a sin ^) 

 ^ /i + a sin 



Multiplying out, this gives 



{ag-bc + bUm6) {h + as\n6) = b^ asin^ 6 + t^ sind (h + asmd), 

 or (aff-bc) (h + asind) = b^asin^d. 



Whence we get 



ha^ — hbc=o; 

 ag = bc. 



(dx\ 2 ^ 



But 



and 

 hence 



/dx + da\^ ( du \^ 



rfar c?a du 

 dt dt dx 



da _ du 

 d t dx 



du _1 da _ 2 ds 

 dx adi s ' dt^ 



whence, by substitution, 



(A + a sin 6) b cos B=2 a cos ^ (c — 6 sin ^) + . , . 



or if we put for h its value found above, 



^J± = 2ae 

 c 



hg = 2e^ 



That is, the square of the velocity of transmission in a triangular channel, is half 

 the square of the velocity in a rectangular channel of the same maximum depth. 



