Variations of the Arbitrary Constants in Elliptic Motion. 265 



&(t$\ Mrs dR T 



and we have —77^ — + ■— y + -i~-> (N), which will also enable us 



to fitad ^ $ see Mec. Cel., Vol. in, p. 17. 



We will conclude this paper by showing how to find the integral 



d 2 u 

 of the equation -jjjf +n 2 w+P=0, (a), where n= const, cfa— const. 



and P= a function of t and given quantities; which will be useful 



for finding the integrals of (H) and (M), which can easily be shown 



to depend on equations of the form of (a). 



Multiplying (a) by cos. nt. dt and taking the integral we get 



cos. nt. du 



-\-nu sin. nt-i-fP cos. nt. dt = a = const., also multiply 



sin. nt. du 



dt 



(a) by sin. nt. dt and take the integral we shall have t: — nu 



cos. nt-\-/P sin. nt. dt=-b = const., then multiply these equations 

 by sin. nt, — cos. nt respectively add the products, and we shall 



have nu=a sm. nt — b cos. n^ + cos. ntfP sin. nt. dt — sin. nt J 9 P cos. 

 nt. dt, (6), which gives the form of the integral as required; see 

 Mec. Cel., Vol. i 3 p. 240, where it is found in a much less simple 

 manner. 



If P = K cos. (mt+e), where K, m, &c. are invariable quantities, 

 we shall easily obtain/'Pcos. nt. dt=Kfcos. (mt+e) cos.nt.dt 



m sin. (mt + e) cos. nt - Kn cos. (mt-\-e) sin. nt r 



■ m *-n* and J P sin# nL 



dt = K/cos. (mt + e) sin. nt. dt 



Kn cos. nt cos. (mt+e) + Km sin. (mt+e) sin. nt 



—z — ~2 ' hence (b) becomes 



m- — n J v ' 



a b Kcos. (mt + e) 

 t*= - sin. nt- - cos. n*+ — ^TZ^ > (c) ; if P=Ksin. (mt+e), 



a b 



we shall have in a similar manner u = - sin. nt — - cos. nt 4- 



n n ■ 



Ksin. ( m<+e) 



If P=K cos. (nt+e), we shall have fP cos. m. df* = It/* cos. 



/ at x t "^ />r . ' /~ v, , Kcos.e.tf 

 V*'+e) cos. n*. d*= — y [cos. e+ cos. (2ntf+e)] eft = -f 



K sin. (2nt+e) 





K 



4- ' also/P sin. nt. dt=--Kfcos. (nt+e) 





2 /[—sin. e-f- sin. (2n*-f e)] <fr 



Vol. XXX.— No. 2. 34 



Ksin.e.f K cos. ( 2nt + q) 

 2 * ~ ~ 4n f 



