Orr — Stahiliiy or Instability of Motions of a Viscous Liquid. 97 



to be a small negative angle. Now, if the argument of a; lies between the 

 limits + TT, we have the equation* 



I-n{x) -Ik(x) 



(9) 



in the sense that, provided s - /v + | is positive, the error in terminating the 

 series on the right after the s'^ term has a modulus less than that of the next 

 term if the argument of x lies between the limits + 7r/2, and less than a 

 certain multiplef of it if the argument of 03 lies between 7r/2 and tt, or between 

 - 7r/2 and - tt. And, by writing in this equation x = ye,""\ and dividing across 

 by sin Im, we obtain the equation 



/_.(7/) + h{3J) - i cot ky [Lify) - h{y)] 



^ ' ^^ I 8y 8.16.f~ j' 



(10) 



which holds in a sense obvious from the preceding sentence, provided the 

 argument of y lies between and 27r. While, by writing in (9), x = ye^""', 

 there results 



Ik{y) + My) + i cot hy {Luijf} - Ik(y)\ 



= (2hv)iey\l + (1-^^)(1 + ^^') + {l-2k){3-2k]{l + 2k]{3 + 2Jc)+... ] 

 ^ ''^'^' \ 8y 8. 16. 3/2 j' 



(11) 

 provided the argument of y lies between and - 2tt. 



Thus, if y is large, and its argument lies between ± 7r/2, it follows from (9) 



that the term involving I.h{y) - Ih{y), which occurs in the left-hand members 



of (10), (11), may be neglected, so that within these limits for large values 



of y we have the approximate equations 



I-k[y) - h{y) = (2/7r2/)l sin k7re-y, (12) 



lAy)-^h[y)h{2iny)hy. (13) 



Accordingly, if AIk(x) + B-k{^) is to vanish for two large values of x, whose 

 arguments lie between ± 7r/2, the values must differ approximately by a multiple 



•a- " On the Product /,»(•*;)/« (-t)," Pi'uo. Camb. Phil. Soc, x., Part III., equations (14), &c. ; 

 " On Divergent Hypergeometric Series," Trans. Camb. Phil. Soc, xvii., Part III., Art. 3, especially 

 foot-notes, pp. 179-180; and Art. 11. In the foot-note on p. 179, for " tt ± 7 " read " + (tt - 7) ". 

 Some en-ata in Art. 11 are corrected in Vol. xix.. Part I., p. 155. 



t The multiplier depends on the argument of x, but not on the modulus. 



