Orr — Stahiliti/ or Instability of Motions of a Viscous Liquid. Ill 



Art. 21. The Approximate Values of the Complex Boots. 



If the point p' lies to the right of the line A'C (fig. 2), the argument of 

 Un lies between - 7r/2 and - 37r/4, so that if ih is large enough, e^'a is small 

 compared with e~"2 ; thus, the period-equation takes the approximate form 



- *e«i + c-«i = 0, (54) 



giving 



«i = {rw + oTrl'^)i, (55) 



where r is zero or any positive integer. This assigns to j/ a position P such 

 that 



f (P^Vv)i//j3 = (7-77 + 377/4) *, (56) 



giving 



|^«-^(|.*-^.)*(../3f |, (57) 



r being any positive integer (including zero), provided r is not so great as to 

 make the coefficient of i negative ; (in that case, we return to the real roots) . 

 A more correct, though still only approximate, equation is that which 

 makes the numerical value of ih satisfy 



Jl\u\-vJ.i\ii\ = Q. (58) 



Equation (58), or its approximate form (56), becomes less and less accurate 

 if the position it assigns for j/ is near C\ as we have seen, p' coincides with G 

 for values of Ui satisfying the equation 



Jl{uii) = 0, or 7^1 = (?'7r + 1177/12)^; 



the r + 3/4 of (55) being thus replaced by r + 11/12, 



It is seen that these values of |)' all lie close to the line CA ; but it may 

 be seen that the correct values cannot actually lie on the line except when 

 at C. And as the roots we have so found, taken along with their images in 

 the axis of real quantities, just equal in number those which have been proved 

 to be complex, all the roots have been accounted for and approximately 

 ascertained. 



Art. 22. In the most Persistent Bisturhance, v is a Function of y only. 



When the wave-lengths in the directions of x and % are increased 

 indefinitely, i.e., when the velocity- component v is made a function of 

 y only, X and I are both zero, and the values of p are given by p = vf-rrlAid' , 

 r being any integer, as may be seen from (15), or, by returning to (1), and 



[15*] 



