154 MB. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



the equation (48) reduces to 



( n + I)* ( n - S ) n* (n + s + 1) n . 



2n-l U "- 1 H 2n + 3 



Thus, if we write for brevity 



N; = w( + 1)-^- 



and put y' = 0, the equation (43) is replaced by the two following : 



(n f !)( - s) n , ,,, ps 



2^n" u - 1 " 



( + I) 2 ( - *) ., S 



- 



1 



(51), 



(52). 



For the determination of the symmetrical types we therefore have the series of 

 equations 



2s + 3 



(s + 2y.l 



2 S +1 



. , 



's + i T 



s+2 ~ ' 



2s +3 



On eliminating the quantities Cj, D' +1) Cj +2 , &c., by means of a continued 

 fraction we find the period-equation in the form 



(53), 



N; +1 - M; +2 - N; +s - . . . ad inf. 



where we have written for brevity a* in place of 



3 (n + 2) 3 (m - s + 1) (n + s + 1) 



(2w 4 1) (2 + 3) 

 In like manner the period-equation for the asymmetrical types may be written 



(54). 



N* _ * * + l *+2 _ f\ 



* ~\/T XT Tl~]r 7 * J "^ " 



(55). 



