58 BULLETIN OF THE 



Gaussian constant, and m the relation of the mass of the disturbed 

 body to that of the sun. 



By putting the first differential co-efficients of the co-ordinates 

 with respect to the time equal to zero, we derive, with great ease, 

 the expressions for the variations of the elements. This is for the 

 case of special perturbations. These expressions will contain the 

 components R, S, and Z. 



If we now substitute the values of these components, wherever 

 they appear, and perform the necessary reductions, we get expres- 

 sions for the variations of the elements, where, instead of the com- 

 ponents of the disturbing force, the force itself appears. 



In the case of the mean anomaly, another method has been fol- 

 lowed. Its variation can best be found by means of the relation 



M=fi(t-T), 

 where M represents the mean anomaly, [j. the mean daily motion, 

 and T the time of perihelion-passage. 



I have thus derived, among others, the equations : 



— _ k (1 + m) JH , -j£ _ k (I + m) d L 



¥ = *'<. + »; !§, 4f = - i! < 1 + ->?#. 



From these, by slight changes, we get the equations used by 

 Delaunay in his theory of the moon's motion. Thus by putting 

 k* (1 + m) Q, = R, and writing I, g, h, for M, w, Q,, respectively, 

 we have 



In these equations, according to the notation of Delaunay, L = 

 l/a~Ji, [i being the sum of the masses of the earth and moon, G = 

 L -j/l — e 2 , H = G cos i ; a, e, and i being the semi-major axis, 

 eccentricity, and inclination respectively ; I designates the mean 

 anomaly, g the angular distance of the ascending node from the 

 perigee, and h the longitude of the ascending node. 



