56 Proceedings of the Roi/al Irish Academy. 



first integral in (29); and, as before, neglecting I^{lir)K^{kci) in its multiplier, 

 the first term of the right-hand member of (29) is replaced by 



—I,{lM)K,{kr)I,{kr)K,{k'b) cos {kz - m^S^ _ l^C't), (69) 



or - -—- . /i {ka)K,{kl) cos {kz - m^lf- - kC't). (70) 



A" 



And in a similar manner it may be shown that if k {a - r) is large the 



second term also of the expression which replaces the right member of (29) 



is equal to (70). Adding, and dividing (29) by 



I^{ka)K,{kh) - I,{kl)K,{ka), 



in which the latter term is negligible, we obtain the approximate result 



It = - 2m'ry¥ . cos (kz - m'h- - m'C'/C). (71) 



But, as in the case of the other disturbance, vj cannot, at this critical time, 



be found from this approximation ; the portion of u which involves the angle 



{on^ + ktC) r"^ - m-¥ - kz + kC't is now more important for the determination 



of 10. 



It may be well to sum up here the suppositions made. They are that 

 k (r ~ h), k {a - r), mr, ni^rlk are each large, and that at the time t, to which 

 these values apply, ni^ - CM = 0. 



A comparison of (71) with (63) shows that, as with the disturbance first 

 instanced, the value of 'u increases very much from the initial one, in a 

 ratio of order m^r-/k- in fact. And, as before, the initial value of vj is 

 much greater than that of u, so that the kinetic energy of the motion 

 relative to the steady motion when averaged along a stream-line exceeds its 

 initial value in a ratio of order m'^r'^/k', assuming that at the critical time 

 w is not of order larger than 7(. 



It would seem that a disturbance of this latter type (63) is more 

 unstable, or less stable, than that of the former type (27), as the critical 

 time in the latter is the same at all points in the pipe ; in fact, the values 

 of the parameters which occur in the approximate equation (71) may be 

 such that the equations are valid through a sufficient thickness of stream 

 to render the kinetic energy of the relative motion through the whole pipe a 

 very large multiple of its initial value. 



If k {a -h) is sufficiently large, equation (71) may indeed be used except 

 through a very small fraction of the thickness of the stream adjacent to the 

 walls. We may in this case obtain an approximate expression for the total 

 relative kinetic energy. For this purpose we may introduce a stream 

 function \p defined by the equations 



ru = dxPJdz, rw = - d\p/dr. 



