28 Proceedings of the Royal Irish Academy. 



smoothly curved, with definite curvature at every point except the vertex 

 This guarantees definiteness of «\ <>'£. 



1". It is useful to test this result by seeing h"\v it applies to some of the 

 particular examples m.'St ^.i-ily available, namely ' -',,. '-,, and ff a as defined 

 in paper D; 55 39 and 40. 



In the case <>f ff tl it is readily seen 'hat o\ is of the order of siuallness 

 of t- lege; but then yi - \_ ". = A' , and so d\ 'II is not definite 

 for £ = ". It is not therefore t" be expected that < ; , should come under 

 the theorem of the pn _ article. 



If ' r : : . tlieL V " -J»|»-2tf/o)*|, and r\. i- 



fonnd to be small of the order of t This is not contrary t" the general 

 theorem; it merely means thai . sucb as to make the coefficient 



■ •I (i zero. This may be couiirmed by sulslituti i the particular form of 



Y £ '" formula 



If «/\ <i% = - A. where A is i constant, the coefficient "f «- is 

 - '\p - 2a\ ir ». which is zero if A r/2a. L'his is just the value of A 



which i esponds to < . ; ; : ind this explains the fact that 



fur this particular cm i Sy. is Bhown by the formulae of D, j 40, to be 



of the order of smallness of t. 



16. The possibility and the importance "i the vanishing of the coefficient 



of t^ in the expansi I r\ render it desirable t<> cany the expansion a 



further by determinii _ lefficient of the term in t. 



T> this end, consider 



n .i 



// I v-'-v - ^ r \< " ■ o - v") r 



■ 



<•-- -s- ' (37) 



noting thai the i definiteness of </\ _ gains) 



divergi : at £ = a. 



By th<- the. rem of the mean the numerator eqi 



-4).{x'lS')-x • * 



■ ■ P ; ind i>t'>0; and this again, by t lie same theorem, equals 

 • - £)c(a - t'- Ox '- 

 win • . > £" > f . H( ace 



I 



J,(a-£)*(a + . - | 





