24 Proceedings of the Royal Irish Academy. 



Aet. 4. Arbitrary Disturbance in uniformly shearing Liquid resolved into a series 

 of Lord Rayleigh's type. Case where initial velocities are sine-cosine 

 functions of coordinates. 



I proceed then, in the simplest possible case, that in which the A^elocity 

 in the steady motion is, from one boundary to the other, a linear function of y, 

 to consider the expansion of an arbitrary function of y in terms of the func- 

 tions which present themselves in Lord Eayleigh's iuvestigation, and to 

 examine the propagation of an arbitrary disturbance. If the fixed boun- 

 daries be denoted by y = 0, y =l>, the problem in expansions is as follows : — 

 Given an arbitrary function /(v/), to find a function 0, such that for values of 

 y between and h 



f(u) = 



i>{n)Ir,{l/)dn (16) 



where Fn(y) is a given function of y defined in the following manner: — 

 ^Hien y is less than rj, F^{y) = A sinh \y, 

 when y is greater than >j, F^{y) = B sinh \[h - y), 



A, B, being connected by the relation 



A sinh \i] =^ B sinh X[i - tj), 



and A being given. As we may take any convenient multiple of each function, 



we will choose 



A = 1/sinh X}}, B = 1/sinh A (& - ??). 



We notice that these functions of y do not conform to the relation which 

 exists among the normal coordinates of a conservative system oscillating about 

 a position of equilibrium, viz. : — 



' ^.A!/)FvAl/)dy = 0. (17) 



With the above values of A, B, equation (16), if it exists, assumes the form 

 By differentiation, we obtain 



'^ ^^^ Jo smhX(b-ri) ]y smhAjj ^ ' 





*(„)coshAy^_^_ (19) 



sinh Ajj 



^ ^, [^ <^(7;)sinhA(5-y) ^^ ^ ^^ p ^sinhA^/^ _ Xcpjy) sinh Xh 

 ^ ^y' L sinhAf&-Tj) '' « sinhAr? sinh A (5- w) sinh 



and hence (20) 



/ m ^JKV) sinhA(5-2/)sinhA2/' 

 giving 



,. ^ \W{y) - r{y)] sinh X{h - y) sinh Xy . 

 ^ ^y' X sinh Xh ' ^"^ ^ 



