$8 Mr. P. F. Ward on the Transverse 



Taking (17) in connexion with (15) we obtain 



A<£ w _iH-Bi|r„_i= 0; .... (18) 



and finally by eliminating A and B we obtain as the 

 equation to determine # l5 and therefore the values of X, 



£(*.-#*-,) = (19) 



To solve (19) we proceed as follows, the method being 

 an extension of that employed by Kirchhoff in the cases 

 of 7i = 0, n = l. If we multiply the two equations (15) 

 respectively and successively by 



y»_i, — <pn-i ; ^ x , c j v ; y n -i, p»-i » 

 and add, we obtain 



* ( ~l^ ^ l_1 + "^^ *^ 1 ) + ^ fc + ^ ^ (^-W^-i) = °> 



which with the identity 

 yield the differential equation 



From (20) it is found that the required series for ^n-i^u-i 



A' 2 a?* 



= 



(w + l) 2 (rc + 2) + 2! ( w +l) 2 (n + 2) s (n + 3)(n + 4) "" ^ 



Therefore from (19) the equation to find X is 



1 x 2 



° = (w + l) 8 (n + 2)""l! (?^+l) 2 (n + 2) 2 (n + 3)(?^^-4) + ■•' , '^ 



which if w is integral may be written 



1 x 2 x 4 



(n+1)! (w + 2)!~"l! (» + 2)! (rc + 4)! + 2! (m + 3) ! (n + 6)!~"" ^ 23 ' 



