flowing doivn an Inclined Plane Bed. 275 



priately in respect to the question of stability or instability, 

 by writing it as follows [£', ? denoting the two roots of (75)], 



and integrating with special consideration of the infinities at 

 y = f and y — %. One way of doing this, which I merely sug- 

 gest at present, and do not follow out for want of time, is to 

 assume 



^=C{f-y + c 2 (r-y) 2 +c 3 ($-*/) 3 +&o.}, 

 + C'{?'-y + c/Gr-y) 2 + c/(?'-y) 3 + &c.} . . (77), 



where C and C are two arbitrary constants, and c 2 , e s , . . . , 

 c <z > G % i • • • coefficients to be determined so as to satisfy the 

 differential equation. This is very easily done ; and when 

 done shows that each series converges for all values of y less, 

 in absolute magnitude, than £' — f, and diverges for values of 

 y exceeding J ' — f. The working out of this in detail would 

 be very interesting, and would constitute the full mathematical 

 treatment of the problem of finding sinuous stream-lines 

 (curves of sines) throughout the space between two " cat's- 

 eye " borders (corresponding to y=f and y=f?) which I pro- 

 posed in a short communication to Section A of the British 

 Association at Swansea, in 1880*, " On a Disturbing Infinity 

 in Lord Rayleigh's solution for Waves in a plane Vortex 

 stratum." It is to be remarked that this disturbing infinity 

 vitiates the seeming proof of stability contained in Lord 

 Rayleigh's equations (56), (57), (58). 



47. Realizing (63), and interpreting the result in con- 

 nexion with (57), we see that 



(a) The solution which we have found consists of a wave- 

 disturbance travelling in any (#, z) direction, of which the 

 propagational velocity in the ^'-direction is —co/m. 



(b) The roots (J, £") of (75) are values of y at places where 

 the velocity of the undisturbed laminar flow is equal to the 

 #- velocity of the wave-disturbance. 



Hence, supposing the bounding-planes to be plastic, and 

 force to be applied to either or both of them so as to pro- 

 duce an infinitesimal undulatory corrugation, according to 

 the formula cos (cot + mx + qz), this surface-action will cause 

 throughout the interior a corresponding infinitesimal wave- 

 motion if co/m is not equal to the value of U for any plane of 



* Of which an abstract is published in ( Nature ' for Novemoer 11, 1880, 

 and in the British Association volume Report for the year. In this ab- 

 stract cancel the statement " is stable," with reference to a certain steady 

 motion described in it. 



T2 



