PROFESSOR KELLAND ON THE THEORY OF WAVES. 511 



, d(t>(xz) dz d(b(xz) „ ._, 



hence -^--- ■ .7-+-^^r — -=^ (7)- 



dz dx ax 



10. To prevent confusion, however, we will at first write do\vn all the terms 

 of equation (6), and afterwards strike out those which occur on both sides, so as 

 to obtain the form of equation (7). 



By (4) - ^= -oi b ccosO (e'V + e-^y) + 2 . b^ asm2e : 



' ^ ' Q dx 



and by (5) -'^ =g'^-^-<^bc<io^6{e'y + g-'^y - e"' - e-"') 



^ ^ dx ^ > dx ■ 



hence -abcfio^Qif-y + e—'y) +262«sm2 



= ^ If _ a 6 c COS0 (e«^ + fi-"!' - £"-- g-" -') 

 dx ^ ' 



^ dx ^ ' dx 



that is, as we should have deduced at once from equation (7), 



X 



i2a= 





2^ 2, 



or 



2rr 25 



^^- 6^ sin 2 e =H^ 6c COS e ( e^" + e" 

 A A 



</-- 27r, . zj / T -T \ Sir ,., / 



d.r r \ \ I X \ 



.^j,-^..sine (;^---.-^->2^.^(>--rT-)| (a). 



Let z=h+ Va-e" +/e '^ / sin i 



2<r 2!r 4!T 4, 



2^, 2 a- 



dt X \ ■' Jdt 



2 cr_ 2 T 



■^-'\2'7r 



^ae^"+/e ^ " ) ^cos 6 ^c- 6 (e '^ ' + e ^ ' ) sin ) : 



But from the circumstance that v =^ when y = z, this g^ves, 



