PROFESSOR KELLAND ON THE THEORY OF WAVES. 



rr - -; - 



i cosup + d a cos a a + o u. 



\L 



_^L (z + Kft/g (z + h) I sin /z p + a a'-sin /zp a + a' 

 + sin /z > + a' -f sin yu a' > 1 c?/>e 



= an integrated function 



If we confine our attention to waves transmitted in the positive direction, 

 this gives 



function 



cos a < 



or if p be greater than a + a', the factor is 



><[' + 3 li 



- 



or oc sn 



-. 



+ & c . 

 J 



_____ 

 o (p a; 



omitting a as a very small factor. 



Hence it appears that the recurring function is independent of a, and conse- 

 quently that a small disturbance in a shallow canal will be transmitted in pre- 

 cisely the same manner, whether the quantity originally disturbed be small or not. 



We cannot extend this memoir to other cases of our present hypothesis, but 

 must pass on to the more general problem. 



72. CASE II. Suppose the expansion to be carried on in terms of ef- *, &c. 

 This approximation is applicable to all conceivable cases of motion, and will con- 

 sequently deserve our careful consideration. 



From equation (9) we have 



d^__bVff r< sin/i( 



dy~ TT Jo ' ' S11< f 1 ' 



e 



