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same set of final integrals, and the same values of #', y 1 , and z', 

 by supposing the elements a, b, </>, ^/, 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 



and let (putting T for nt+p), 



a cos cos T b sin sin T =, 



a sin (p cos T + b cos sin T= / ; 

 then 



S(nab)=a cos T(P y ) 6 sin 

 a* + !>*))= -n(a sin T(P,)- cos T(P y )) + rW 



cos L ty)=^(b cosT(P x )-a sin T(P y )) + 2a b sin 



It may be observed that and TJ 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 



#" + w 2 O-X)=0, &c., 



where w 2 , X, Y, and Z are any variables, may be solved by the same 

 set of final integrals, and the same values of #', y', and z\ as those 

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

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



