m 



Dr. Hargreave on the Problem of Three Bodies. 469 



then —fn^s.dl, 



nab sin i c^=(Z')-t-, 



^(K«6) = -«((X') b cosT + (Y') «siuT), 

 ia(wX«' + &'))= -»'((X') a cos T + (Y') h sin T) + r-nn', 



{a^-h') {i<p+ cos I c;P)= -((X') 6 sin T + (Y') a cos T) + 2a 6 sin T cos T^p 

 (a--i^) a(?i; + p)=(X') GsinT + (Y') Sees T-(aH^'') sin Tecs T^; 



in which ~ and -tt are the differential coefficients of the expressions 

 at at 



for 'i and >/, taken explicitly with regard to t. 



This method is denominated the method of Osculating Variation. 



Applying the method of tangential variation to the system 



a." + iia'=0, &c,, 



we perceive that this system admits of complete solution in finite 

 terms, leading in fact to the usual theory of elliptical motion. Taking 

 this system, therefore, as a normal system, the author proceeds to 

 deduce the formulae for the variation of the elements of this system, 

 iu order to arrive at the solution of the system 



a;" + iia'=P,,, &c. 



The elements which have been selected, for reasons fully explained 

 in the paper, are t and »//, wiiose meanings are already known ; A 

 and Nr denoting respectively the mean distance, and the longitude 

 of the epoch measured in the plane of the tangential ellipse as it 

 exists at the time t, and measured from the node at that time ; and 

 e and is denoting respectively the eccentricity of the tangential 

 ellipse, and the longitude of its perihelion measured as above ; and 

 it is observed that these are strictly normal elements, according to 

 Pi'ofessor Donkin's definition of normal elements. 



The variations of these elements are then rigorously found, and 

 are expressed as follows : — Denote 



cos \^ Vx + sin i// Vy by the symbol Pf, 

 and 



cos I (cos (/'P.i — sin i/zPj,) + P~ sin t by tlie symbol P, ; 

 and let 



— Pj sin + P, cos = Pf ^9 ; — P| sin CT + P,, cos ra- = P{,„ ; 



Pf cos + P,,sin0=P,,_g ; Pj cosra^ + F, sinOT=:P^,„ ; 

 then s r cos 9 ,^ ^ 



^'=NAXl-.^)^^^~'^' 

 „ , r sin 6> ^„ • 



'"^""^T.NAX1-.^)*^^^^' 



I 



