I 



468 Royal Society :— 



where tr and P,,., P,/, and P- arc any variables, may be solved by the 

 same set of final integrals, and the same values of x', y', and s', 

 by sap])osing the elements a, b, «/>, 4/, t, and p to become variable. 

 These elements are those of an ellipse tangential to the actual 

 curve of motion ; and the following formulae are obtained for their 

 variation : — 

 Let 



\a^,+\oVy +X,P^=(P,,,), 



and let (put ting T for nt+p), 



a cos cos T — b sin ^ sin T= ^, 



a sin cos T + i cos ^ sin T=); ; 

 then 



nabli=l[,(P.), 

 nab sin i ^;|/=>;(P.), 



l{nab)=a cos T!{7y)-b sin T(P,,,), 

 \l{rr{(r + L')) - -n{a sin T(P,) - b cos T(P2,)) + rnn\ 



\ ?i' 



(«-- 6=) (<> + cos I U)=-(b cos T(P,,)-« sin T(Py)) + 2ab sin T cos T-, 



{a'-b') g(«^ + p)= -i(« cos T(P,,)-i sin T(P^))-(«- + J') sin T cosT^' 



It may be observed that ^ and »; are coordinates of the body referred 

 to the plane of the tangential ellipse, and to an axis of ^ coinciding 

 with the node. 



This method is denominated the method of Tangential Variation ; 

 and it is applied directly to the problem of the circular pendulum, 

 that of the spherical pendulum, and that of the motion of a particle 

 where the force is a function of the distance, and in particular that 

 of elliptical motion, where the law of force is that of the inverse 

 square. 



In a subsequent part of the paper it is shown that a system of 

 the form 



x" + 7iXa; — X) = 0, &c., 

 where n', X, Y, and Z are any variables, may be solved by the same 

 set of final integrals, and the same values of x', y', and c', as those 

 which have been already given as the solutions of the same system 

 when 71, X, Y, and Z are constant, by supposing the elements to 

 become variable. In such a case, the elements are those of an 

 ellipse osculating with the actual curve of motion, always of course 

 having its centre at the moveable point (XYZ). The following 

 formulae are obtained for the variation of these elements : — 



Let 



X„X' + A6Y' + XeZ' = (X'), 



/.„X' + ^«Y'+;u,Z' = (Y'), 



,'„X' + .'4Y' + r,Z'=(Z'); 



