334 Lord Rayleigh on Stability of Simple Shearing 



iuncticms fluctuate with great rapidity and with correspond- 

 ingly increasing maxima and minima. When in one period 

 ^/2.7] s ^ 2 increases by 2tt, the exponential factor is multiplied 

 by e 27r viz. 535'4. From the approximate expressions appli- 

 cable when 7] exceeds a small integer it appears that s 1? t x 

 -are in quadrature, as also s 2 , t 2 . 



For some purposes it may be more convenient to take 

 .% u 2 2 , or (expressed more correctly) the functions which 

 identify themselves with Si, 2 2 when 77 is great, rather than 

 *Si, S 2 , as fundamental solutions. When 97 is small, these 

 functions must be calculated from the ascending series 

 Thus by (15) (0 = 1, D = 0) 



5 1 = 7r-^r0S x -37r-*r^S 2 , . . . (25) 

 -and (0=0, D = l) 



5 2 = 7r^rQ^-^6S 1 + 37r^r(?) ^/ 6 S 2 . (26) 



Some general properties of the solutions of (5) are worthy 

 *of notice. If $ = s-tit, we have 



dh/drf = 977*, dH/drf = - 9 V s. 



LetR = -|(s 2 + * 2 ); then 



dR,_ ds dt 

 dt] dt) drf 



-and d 2 R (ds \ 2 (dt\* dh_ dH^ 



dv 2 ~\d V ) + W **di? + d V 2 ' 



of which the two last terms cancel, so that d 2 R/d7j 2 is always 

 positive. In the case of Si, when 77 = 0, ^(0) = 1, £ t (0)=0, 

 5l '(0) =0, so that R(0)=i, R'(0) = 0. Again, when v = 0, 

 .j 2 (0) = 0, * s (0)=0, so that R(0)=0, R'(0) = 0. In neither 

 case can R vanish for a finite (real) value of 17, and the same 

 is true of S x and S 2 . 



Since (5) is a differential equation of* the second order, 

 its solutions are connected in a well known manner. Thus 



s ^- 8 w= ' • • • • (27) 



.and on integration 



S^ -&2^f = constant =i, . . . (28) 



