GENERAL INTEGRALS OF PLANETARY MOTION. 31 



variations of the orbits are so rapid that the approximation in powers of the time 

 fails even for present uses. Hence, the lunar theory, considered as a problem of 

 three bodies only, is always treated in a manner analogous to that in which the 

 general theory of planetary motion has been considered in the present paper, the 

 three arguments introduced by the moon being her mean longitude, and the longi- 

 tudes of her node and perigee. In the theory of Delaunay the analogy in question 

 is most easily seen. His Z, G, H, represent three of our canonical elements c^, 

 the constant term of R, to which he constantly approximates, is the constant part 

 of so much of the expression for the living force as contains L, G, and H, by differ- 

 entiating which with respect to the latter quantities, he obtains the expressions for 

 the motions of the three arguments. 



The theory of Jupiter's satellites has been treated by M. Souillart in such a 

 manner that the co-ordinates may contain, instead of the longitudes of the peri- 

 ioves, the varying angles on which these longitudes depend. His analytical theory 

 is given in the Annales de VEcole Normale Swperieure, Vol. 2, 1865. 



It may be hoped that the general view of the subject taken in the present paper 

 will afford a means of introducing a more rigorous system of integration in such 

 cases. One of the special problems growing out of this general theory will be the 

 determination of the coefficients of the time, b-^, \, etc., either in terms of the canoni- 

 cal constants Ci, Ca, etc., or of the largest of the coefficients Ic, in the expressions for 

 the co-ordinates of the several planets. These coefficients are, approximately, the 

 mean distances of the planets. The quantities h ought, perhaps, to appear as the 

 roots of an equation of the 'inth degree, but the writer has not yet succeeded in 

 forming any expression fitted to give rise to such an equation, except one in which 

 only the squares of the quantities in question appear. 



PUBLISHED BY THE SMITHSONIAN INSTITUTION, 

 WASHINGTON CITY, 



DECEMBEH, 1874. 



