104 Mr. P. F. Ward on the Transverse 



Consider the case o£ a canal of rectangular section, and let 

 the depth h and the breadth b vary as w m and x n respectively. 

 It 7*o and b be the values of h and b at x = a, we have 



7_7 (A m 7 7 M" 



a) 



and therefore (2) reduces to 



where k 2 = <r 2 a m /gh , 



or ^j"! + (m + n)x m - 1 ~ ■ +/e 2 77 = 0. . . . (4) 



9 2 ~ TO 

 Writing x 2 = f, equation (4) reduces to 



2—m 





where 2p = -= . If we now write r) = tj^ p 2 the equation 



finally reduces to 



<*P pf I p~ J '' " " () 



which is a Bessel's equation of order (p— i) or 

 (?n + 7i — 1)/(2 — 7?i) . 



In the case of 771=-. 2, the above transformations fail, but 

 equation (3), which then becomes 



at! 12 7 a# 



is readily solved, being a homogeneous equation. It may 

 be noticed, however, that when m = 2 the solution becomes 

 infinite for # = 0, and therefore the origin must be excluded 

 by means of a barrier from the stretch of canal considered. 



Returning to (5), since 7?i and 71 are both necessarily 

 positive, and the solution has to be finite at the origin, a 

 complete solution for our purposes is 



r 2 



or y = - s+SZi J m+n-i (g-Z^ g a J' • • • ( 6 ) 



,77 2 2-ot 



