74 Proceedings of the Royal Irish Academy. 



n - i ^- >■ ~ Mm Ki (Lrn - ?/*■•£■ + M.-K".) , K",- (Z.-V,- - 2/1,-^Z.- + /x.-^ A,) | 

 ,, _ i &.Qu,- - ju^) Z, (Z,--/y - 2,<,Z, + ftjKi) LijLjh-i - 2 f i ! . i r. i + ju^A,) ! 



Z\ = | ftfo - M..0 -T 3 - A',Z, (A".-7/*r'V - Kt'fftfrdh 



"'. for the sake of clearness, even at the expense of brevity, as little 

 change of notation as possible has been admitted up to this point, it is now 

 necessary to make a transformation which will allow two sections of the 

 characteristic function to be fitted together. Let 



K II q K '</,.! = //!>),■-,, L,",'/:' = Him, /-..".'-i = ffi'vi'-x 



with the same forms obtained by substituting i s, Z'i Eor //, i|). The succeeding 

 refraction cat ited in exactly the same way. by changing the subscript 



i ini" i 1. The terms of the second order in JV itl are removed and Win" is 

 obtained by substitution. But consistently with the above transformation 



A .,■ //, M 7,. /-,,,.'/,' ■= Hi'^m', 

 ami this requires that 



//. : IL A'.., A. // . Hi I. -., Lr. 



u : <Z,-A', K /. u < I. l\.. l )/K i \ 1 L i ' n , 

 since the second pair ••!" plan become the first pair for i + 1. Also if 



////• v Li-Kt), 



whi rly consistent with what ; ecome 



u -,w 



7.'-i 



' * 





y, y. " y y I " [ ' 



- - - 



(10) 



- 1 ' 



Let 



p,' = „,* v r. - - aw// „ , = ,/r., 2 + s.-! 2 = /w//, 2 . 



-, = Vi-i'"' + &-i&' = 'I)/ /I, J/, 

 and al 



//. = A,,-, Z, - A',), //,' - ''/'•, A, - K . 

 Thus 



A.A.' = // // /-. - A'.) 2 = Z, - A,)" 1 



I A i _ ^ /- 

 h, A | /•,., /. ., A ,.,) /", Z ,., 



/(,'., _ Z,' M /- A i _ /..J AV 



hi Li' I A ry A'^j 



