92 



Mr. E. R. Darnley on the Transverse Vibrations 



but the terms of the second degree are of the form of 

 (cf>r 2 — ^r 2 ), and when K/r— >7r the terms of the second 

 degree in cosec KZ r are of the form cot 2 K/ r — cosec 2 K/ r or 

 unity. 



Hence, when Kl r — >7r, /\ n is of the first degree in 

 cosec K/j. and /(K) remains finite. /(K) is thus a continuous 

 function without poles. 



Let KZ = 7r where nl=l l + l 2 J r ... + l n , and denote by 

 /(K)^ the value of /(K) when K/ 1; KZ 2 , ... KZ„ all tend to 77- , 

 and by/(K) the value of /(K) when k tends to zero. Then 

 if /(K)„. and/(K) have opposite signs, the equation /(K) =0 



will have a root between -j and zero. 



Let KZj = it + a l5 K / 2 = 7r + a 2 » • • • K Z„ = it + a w , 



where a 1} a 2 . . . a n are small and a l -\- a 2 + ... + a ?l = 0. 

 Let cosh7r = «r, cosech ir=y. 

 Then, as far as the second order of small quantities, 



f(K) ll = a l , a 2 ... 



2, 



1 



*+- 



a 1 



1 



+ 



a 2 



1 



— 



1 



a,' 

 1 





o, 



1 



y 



a 2 



2a 



• + 



a 2 



+ 



«l' 



3/ 



«s 



1 •, 1 1 



y ,2x-\ 1 , 



a 3 a s a± 



1,11 



— , 2*4- + - 



a n _i ctn-i a? 



Denote by t D„ the value of this determinant if a? and y were 

 zero, and by r D s the value of the similar determinants where 

 a s are retained. Then by a property of 



only a r 



'r+l? 



determinants 



,D«= 



a r + a r+ i+ .. . +a s 



The terms of the first order of small quantities in /(K) ff 



are a 1 a 2 ...a n • iD n , and vanish since a 1 -\-a 2 -\- ... +a n = 0. 



To find the terms of the second order denote summation 



1 

 from r=p to r = q (p%q) by X- 



The determinant may be expanded by Laplace's method 

 in a manner analogous to that given by Muir for continuants 

 (Proc. Roy. Soc. Edinb. 1874, p. 230). 



