88 Proceedings of the Royal Irish Academy. 



value given in (22) if the real parts of a, h are both positive, but not if either 

 or both are negative or zero. 



A system subject to an equation of the type 



dxjdt + ax = X 



affords a still simpler illustration, and might be held to be more appropriate 

 to the problem in view. 



Aet. 9. Other Objections to the specicd Solution as a Proof of Stahility. 



The same penultimate sentence of Lord Kelvin's investigation also contains 

 another unproven assumption, viz. : that i) comes asymptotically to zero as t 

 increases to oo . This statement, like the preceding one {i.e., that it rises 

 gradually from zero at ;J = 0), is only known to be true for the boundary 

 values of i). This objection to the second statement may be expressed as 

 follows : — In the first place, the fact that the value of v, simply-periodic in 

 time which satisfies (2), (3), (16), (17), can be expanded in a convergent series 

 of powers of y, does not preclude the impossibility of so choosing w, I, m, n, 

 that V could, through some portion of the interior, be made very great, or 

 even as great as we please, compared with its values at the boundaries ;* and 

 in the second, the mere fact that the resultant value of D is obtained as the 

 integral effect of such solutions corresponding to different values of w, when 

 viewed in the light of the known possibilities of Fourier analysis, so far from 

 showing that it eventually diminishes indefinitely, is seen to impose no limit 

 whatever on its value. 



Again, the tacit assumption that, if the steady motion is stable for distur- 

 bances in which v varies as sin my, it is also stable for those of a more general 

 type, appears to require justification. 



Art. 10. The Sjjecial Solution contains a Proof that the Motion, if ra-pid enough, 

 will he practically Unstalle. Two Modifications of the Solution 

 partially satisfying the Boundary-Conditions. 



Thus, Lord Kelvin's special solution, equally with that included in his 

 discussion of the more difficult problem, appears unacceptable as a proof of 

 the stability of the steady motion. We have seen, however, that if it be 

 admitted, as will be proved in Chap. II. below, that the infinitesimal prin- 

 cipal disturbances have stability of the ordinary simple exponential tyjDC, 

 it does provide an investigation of the propagation of an arbitrary initial 



* It may be held that this remark, if it stood alone, would not affect Lord Kelvin's inference that 

 the sleadjf motion is stable if the initial disturbance be of the tj^pe he chooses and siifficienthj small. 



