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XIX. On the Determination of the Numerical Values of the Coef- 

 ficients in any series consisting of Sines and Cosines of Multiples 

 of a variable angle. By Sir J. W. Lubbock, Bart., F.R.S., fyc* 



IN the following methods of developing any function of sines and 

 cosines by means of particular values of the function, the artifice 

 by which the coefficients are successively eliminated is taken from 

 Le Verrier^s " Developpements sur plusieurs points de la Theorie des 

 Perturbations des Planetes," No. I.f That illustrious philosopher 

 has not yet given the values of the numerical constants which are 

 requisite when the method is practically employed. 



Let 



J2 =.B +£ 1 +£ 2 +i?3+&c. 



S 1 = B +B 1 cos l+B 2 cos 2+B 3 cos 3+&c. 



R^ — Rq-^-A^ sin 1+^2 sm 2-\-A 3 sin3-\-&c. 



R 2 =B +B 1 cos 2+B 2 cos 4+B 3 cos 6+&C 



-\-A x sin 2-f A 2 sin 4-\-A 3 sin 6-(-&c. 

 R 3 =B +B 1 cos 3+J5 2 cos 6+B 3 cos 8+&c. 



-\-A x sin 3-j-^ 2 sm 6+^3 sin 8-f-&c. 



Let Bj=Bi cos i + A { sin i 



R j+ , = B { cos (t + At) + A t sin (i + At) 

 Bj^—Bi cos (i + 2 At) + ^ sin (t + 2 At) 

 -B> + Rj+2 — 2 22y+ 1 cos 

 ss JS^cos t + cos (t + 2 At) — 2 cos (t + At) cos 0} 

 + ^{sint + sin(t + 2 At)— 2 sin(t-f-At) sin 0} 

 = 2 B t cos (t+ At) {cos At — cos 0} + 2 ^ sin (t + At) {cos At— cos 0} 

 = {2B ( cos (t + At) + 2^ sin (t+ At)}{cos At— cos0} 



= — 2 2 { B { cos (t + At) + ^ sin (t + At) } sin - (At + 0) sin - (At — 0) . 



If this expression be compared with the expression for R J} it will 

 be seen that one of the new factors of B i} which I call the variable 

 factor, is cos (t + At), the coefficient of B { in the expression for 

 R j+i ; the other factor, which I call the constant factor, 



cos At— cos 0, 



is independent of i ; and if be assumed equal to A„ B { and A t are 



eliminated, because sin - (A i — 0)=O. 



* Communicated by the Author. 



t A translation of this memoir appeared in Tavlor's Scientific Memoirs, Part XVIII. 

 p. 334. 



