92 Proceedings of the Royal Irish Acaclemi/. 



was rotated in water at 16"7' C. outside a concentric one of radius 14-3930 cm., 

 the motion ceased to be thorougiilj staljle when the speed exceeded aljout 

 56 revolutions per minute; taking v to be "Oil, this corresponds to a value 

 of l^'lv which is aliout 1940. Writing /3&Vi' = 1900, it is seen that the 

 disturbance could not increase greatly. Going back to (34), but writing 

 m - Jftt = 0, and retaining only the terms which are more important, we have 



T 



Y'- 



. ( -25^m(3/^ + 7/r) | P + m^ ( tanhi^j 



The final factor is less than unity, and also less than I-b-/12 ; thus its 

 value is less than 



and also less than 



^^^ \t^J^ 3/-^//-+ rnW]\ . ,, • (39) 



. 5700/5 V J\ 24 



For either of these expressions to be a maximum, there is requu-ed 



[m-h' + r-F-y- = 1900Ih.mh, (40) 



or, if m/l is supposed large 



m'¥ = 1900/5 ; (41) 



then the former becomes approximately 



1 _^ (190Q)- \ 

 and the latter 



(42) 



24 "24Tl900?J ("^^ 



A superior limit to (37) is thus the smaller of (42), (43), and thus their 

 common value, when they are ec^ual, i.e., about 15. The maximum value 

 of (87) appears in fact to be about 4 ; and it approaclies this value when 

 /5 = 2, mh = Btt. 



It may be seen that, for this value of ^h'jv, the terms omitted from (34) 

 are imimportant, and that the approximations used give nearly its maximum 

 value and the time at wliich that occurs. 



If the disturbance were taken alone which involves the first exponential 

 factor in (31), (32), somewhat similar results would be obtained as to the 

 possibilities of its increase. 



