PHILOSOPHICAL SOCIETY OF WASHINGTON. 57 



A communication by Mr. W. F. McK. Ritter was then read, 

 entitled — 



ON A SIMPLE METHOD OF DERIVING SOME EQUATIONS USED IN 

 THE THEORY OF THE MOON AND OF THE PLANETS. 



The rectangular and polar co-ordinates of a heavenly body are 

 functions of the elements of the orbit and of the time. When the 

 elements are pure constants, as in the case of undisturbed motion, 

 these co-ordinates vary only with the time ; but when the effect of 

 the disturbing force is considered, we have variation or perturba- 

 tion of the elements, and hence, also, the co-ordinates vary both 

 with the time and the elements. 



Since the co-ordinates are functions of the elements, as long as 

 the variations of the elements are unknown, the corresponding cor- 

 rections to the co-ordinates, due to these variations, must be re- 

 garded as zero. Hence, in the differentiation, the differentials of 

 the co-ordinates with respect to the elements, alone considered as 

 variable, must be put equal to zero. Hence, also, the velocities of 

 the rectangular and polar co-ordinates are zero, and thus we are 

 furnished with equations of condition, which greatly facilitate the 

 solution of the problem of determining the perturbations of the 

 elements. 



In finding what are called the special perturbations, we resolve 

 the disturbing force into three components. 



For this purpose, call 



E,, the component in the direction of the radius-vector, 

 S, the component perpendicular to the radius-vector, parallel 

 to the plane of the orbit, and positive in the direction of the 

 motion, and 

 Z, the component perpendicular to the plane of the orbit. 



The values of these components, in the form we wish to employ, 

 are 



K = A« (1 + m) 13, 

 S = £ 2 (1 + m) - d J^, 



Z = # (1 + m) 2JS 



Here Q> is the disturbing function, r and v are polar co-ordinates, 

 2 the* co-ordinate perpendicular to the plane of the orbit, k 2 the 



