58 Proceed mg8 of the Royal Irish Academy. 



made indefinitely nearly unity, we revert to the problem of the preceding 

 Chapter for the case in which, in the notation of that Chapter, Ih is large, and 

 it is easily verified that then equation (71) above agrees with (38) of 

 Chapter I., and that the results deduced for the values of T at the critical 

 time agree also. 



Art. 21. The Bisturlance of Art. 20 increases greatly for other Relative Values 



of the Constants. 



In the case of the disturbance of type (63) the approximate values of 

 the velocities at the critical time may be obtained and similar conclusions 

 drawn for other relative values of the parameters than those stated above. 



Suppose, instead, that mr, mh-jk are large as before, but ka, and therefore, 

 of course, also kr, kh small. We now use the approximate values of the /, K 

 functions appropriate to small values of the argument, viz. : 



I,(x) = xl2, K,{o:) = llx. (79) 



Considering the first term of (65), it is even easier than in the former 

 case to prove that the most important term in it is that involving tit^p^ ; and 

 evidently at the time when nv^ - ktC is zero it is approximately equal to 



m'h-' cos [kz - onW - nfC'/C} [' p'(p' - 1% (80) 



that is, to m' {Zr' - Wr^ + 2&0/155 . cos ( kz - m^W - m^CjC ] , (81) 



or to m'(r - iy{3r' + 6r'h + 4:r¥ + 2h')/15h . cos { kz - m^V - iri'C'IC] ; (82) 

 and at a point whose distances from the boundaries have a ratio neither very 

 large nor very small, this is of order m^ {a - lif{a + hyih, provided, of course, 

 the cosine is not a very small fraction. 



It may be proved also, that under these conditions, the second term of (65) 

 is negligible in comparison with (82). Consider first the portion of this 

 second term which involves m^p^ ; on substituting the approximate values of 

 the /, K functions, and the critical value of the time, it is seen to be nearly 

 equal to 



- m'h-' \ p- ip' - ¥) cos (27>z>^ - rii'V - kz + kC't) dp, (83} 



or, integrating by parts, to 



- \m^h''^ [r^ - h^r] sin (2^1^^ - m^b'^ - kz + kC't), 



+ Im^h-' ['' (3^- - ¥) sm (2»rp^ - m^h'' - kz + kC't) dp. (84) 



At a point such as referred to, the first term of this is of the order 

 m^ {a - h){a + hyi'^, and therefore negligible compared with (82), provided 



