70 Proceedings of the Royal Irish Academy. 



This expression is to be transformed in the light of the first approxima- 

 tion (6). Now, 



I A" dfi-vO- L{ < j/,V 



= Wyi. - L' . • - - -1 (2 - . /. '/ - /. 



= {LCul-UvL^- - ■' -'- K"vi-Xi'y^ W K 



- KTfi-K, : K - ■'■ -L /. • . 1 -KiU'lUK ' 



l-Z/AV A'. V., "• -/..V.vl- KlLl'lUR 



But tlie tirst approximation shows that 



K 



are of the third order : and 



A',7. - /.A' A,.-, - „,) - i /., •, - „, ... A"-, - „,i 



A", . 



II . the sixth order being neglected, 



" / ' /V 

 \ I A. /Y, ' 



- i / . . . (8) 



he other hand, the transforn>ati"ii (7 applied to (1 gives for the 

 I • 



Sir : . ■ , k i. 



- /. K | f I A',' A," E 



. . .. /. K A , /. cy/|. 



H( :. •" .ly, 



I V 1 /. /. 8y/ 



..' ; - A',':,., / 

 and therefore 



r ■ r- • {L, - Kj) dW? 



r 



L. A". .- - AV- . - -■ ,/ "- A " 



- 



... 



