Manchester Metnoirs, Vol. xlvi. (1902), No. 15. 9 



In order to obtain an expression of the form 

 e'""^ cosinU, we must combine our functions as 

 [cos;//( U-{- iV) + cosw( U- iV)] 



+ /[sinw( 1/+ iV) - sinw( U- iV)]. 

 To see in what cases Lord Rayleigh's result holds good, 

 we notice that, from (7) and (10), 



a 4tanh/3tanhy ai 

 i>~ {i +tanh-y)- bi 



and as the combination now considered is included in the 

 assumption 



a _ai 



we deduce for it the condition 



4tanh/3tanhy = (i +tanh2y)- . . . . (13). 

 On introducing //, k, and g in place of j3 and 7, this 

 gives 



kY - WkY + U\2,^' - 2K')q' - I (yhXk"^ - /^^) = . (14), 



which is, mutatis mutandis, identical with the bicubic in 

 the paper referred to above. 



To this result we may give a somewhat simpler 

 appearance by writing ^-p, or cosh^ — sinh^, for tanh-y, 

 and therefore cosh-^^-?" for tanh/3. Then 



^ _ i-g-^^ ^ I - coshyg- ^P ^ (coshV-i)e-^^ 



whence 



,^ = ' =(3 + cosh20)(i -cosh2^ + sinh2^) . (15), 



from which 20, and therefore q, can be found by the aid of 

 Tables, by trial and error and subsequent approximation. 

 I now revert to the consideration of a single pole, to 

 examine whether a simple case in which discontinuity in 

 the stresses presents itself can still be made to satisfy the 

 conditions of the problem. 



