86 Mr. P. F. Ward on the Iransverse 



Assuming %=ucos\t, this becomes 



pcoX 2 d 2 ( d 2 u\ , . . 



T^srs?) () 



Now let the dimensions parallel to x and y vary as z m and 

 z n respectively. Then, the section being rectangular, we 

 have 



co = a>'z m + n , k 2 = K ' 2 Z 2n , 



where o>' and k' are the values of o> and k at z = l. Sub- 

 stituting these values in (2), we have 



• o\ 2 d 2 / d 2 



E^ 



ch 2 V 5?/ ■ ■ • W 



The solution of this equation in terms of infinite series is 

 given by Kirchhoff. We need only notice that if m > 2 the 

 analysis breaks down at the point s = 0, the solution becoming 

 infinite at this point. 



The most interesting cases arise when m = l. The equa- 

 tion (3) now reduces to 



F t K ,2 ~ dz 2 y dz 2 r • • • • w 



which may be written 



Hence we have to solve the two equations 



*h-i&* dz~ ± VjM* u - ' ' ' W 

 Writing 2\ / (™~t 2 ) = *» these reduce to 



l£ we now put 4# = ?7 2 , we have 



e? 2 ?/ (2n + 3)dw 



^ + -T~^ =±W ' ■••■(*) 



* The substitution here used is a special case of the more general one, 



