30 Proceedings of the Royal Irish Academy. 



which might be prescribed as alternatives), the results obtained afford a 

 satisfactory explanation of the instability which is observed. Consider first 

 the value of v as given in (28), without regard to end-conditions. The 

 presence of the expressions 



X^ + (m-p)^ X'+(m + p)^ 



in the denominators shows, indeed, that v eventually diminishes indefinitely ; 

 but the occurrence of the former shows that before doing so the disturbance 

 may increase, and increase in a very great ratio, or rather, as Professor Love 

 has reminded me, that it may increase to such an extent that the equations 

 (29), in which as usual only the first powers of u, v, w are retained, may cease 

 to fairly represent the motion. If m is large compared with X, i.e., with (Z^+w^)*, 

 then, as t approaches a value T given by m - l(3T= 0, the second fraction in 

 the right member of (28) becomes negligible compared with the first. At this 

 particular instant of time the first term gives for v the approximate value 



X' + m^ sinh \h - sinh X(h- ?/) - sinh \y . ^ 



V = ,. , B . ,\, sm Ix cos nz 



2X^ sinh Xb 



+ m' 



^ ( ^ cosh X (y -ib)) . , 



^{/- cosh 1X5 h^^^^^^^^^- (42) 



■ 2X^ 



The average value of this between the limits y = 0,y = h is 



\^ + m} ^ ( . tanh4X5) . , 



^ { 1 r^4 — ) smfo cos%5!. 



2X^ ( |X5 



If X/m and mb are each large, the ratio of this to the average initial value of 

 V is great whatever be the value of X5. In the extreme case, in which Xb is 

 very great, the ratio is approximately 7rmV4X^ ; in the other extreme case, in 

 which Xb is very small, the ratio is approximately Trm^b^/AS. 



In the two-dimensioned problem it may be seen that the average value 

 of u does not increase in so great a ratio as that of v. It should be noted, 

 moreover, that at the critical time when m - IjSt = the most important 

 part of u may be contributed by the second fraction in the right-hand member 

 of (38), instead of by the first. In estimating the extent to which the dis- 

 turbance as a whole is increased it must be borne in mind that, if m is large 

 compared with /, the original value of v is small compared with that of u, so 

 that the kinetic energy of the relative motion does not increase in so great a 

 ratio as does ly' ; it appears, in fact, that this energy increases in a ratio which 

 is of order m^/P if lb is large, and of order m^b- if lb is small. This follows 

 most easily by using the stream-function. The velocity-components w, v are 



