138 Proceedings of the Royal Irish Academy. 



in which term.s involving k^ have been neglected. We have then to choose a^ 

 so that this value shall be least possible. The requisite value of oi is not well 

 defined, but is in the neighbourhood of 3'7. Substitutions of ai^ = 3"5, 3'7, 4 in 

 (93) give respectively - k'^ai^ = 1940, 1938, 1 946. In these, however, the terms 

 involving /fc* in (92j have been neglected. If we substitute the approximate 

 values just found in three terms of that order which are given in (92), and 

 take oi^ = 3'7, we now obtain - Pai" = 2027, 1/10 of the increase being due 

 to the last of the three terms. With this value we finally obtain 



DWpl^ = C'ct'pl^ = - 4^^^«l^ = 180. 



It appears that we may safely neglect terms in which higher powers of k than 

 the fourth occur ; the term involving k^ which is given in (92) is presumably 

 the most important of these ; and on substitution of the numbers just found, 

 its value is seen to be about 1/1000, 



