38° J. H. ROHRS, ESQ. ON THE 



Then if /(*) be expanded in a Fourier's series of cosines of the form 



n being any positive integer, f (a), /"(«), /'"(«) obtained by differentiating the series for 

 /(*). w iH be the same as if obtained by direct expansion in a sum of definite integrals. 



For //(*) cos [ j da 



./(*) sin — — - // s . sin — da 



(2n + l)ir 2 a (2« + l)7r J 2 a 



2a .. . . 2ra + 1 tts 4«r . 2» + 1 tts 



; ; — . t (a) sin h r « cos — 



(*» + l)ir /w 2 a (2» + l)V / 2 a 



4a 2 2n + 1 tts , 



~ t ct - ; 7/ • cos as 



(2« + 1) 2 tt 2 •'• / 2 a 



2a „, . . 2w + 1 it* 



./(«) sin 



(2n + i)tt 2 a 



4a* 2w + 1 tts 8ffl :l . 2»+l7r* 



^— -^/ S cos^_-^-— _ ./ (,)«„ __ 



8 « 3 ,-..,„, > . 2W + 1 7T 



+ 



7- rr— ; ff'"(s) sin ds. 



Taking the integrals between the limits a = and a = a we find in all these three equa- 

 tions, that all the terms on the right hand side of the equation vanish, with the exception of the 

 last term ; hence calling the n th coefficient in the direct expansions of /(s), /'(s), /"(a), /"'(«), 

 A n , A' n , A" n , A'" n , respectively, we have 



A =- 2a a> A *"' A „ 8« 3 > 



which proves the property in question. 



It is quite possible that even if/'(s) vanishes when a = a, and/(s) did not vanish when 



0M X 1 77- 9 



s = a, that /'(s) could be expanded in a series of sines of — , with advantage of 



2 a 



calculation, just as Professor Stokes has shewn, that f(a) can be expanded in a series of sines 

 of 1 7r — ) when /($) does not vanish, when a = a, frequently with gain in labour of numerical 



details; but in the cases in which I have employed 2 An sin to represent /(*), 



2 a 



/(«) has always had its greatest value when a = a. I shall proceed now to form the partial 



differential equation for the oscillations of a flexible chain disturbed only in a vertical plane, 



small quantities of the second and higher orders being at first neglected. 



