196 
MR. IVORY ON THE THEORY 
both in brevity and clearness, will be obtained by adapting the investigation 
to the particular circumstance of the case, and attending solely to the princi- 
ples of the method in deducing the solution. It may therefore become a ques- 
tion whether it be not possible to simplify physical astronomy by calling in 
the aid only of the usual principles of dynamics, and by setting aside every 
formula or equation not absolutely necessary for arriving at the final results. 
The utility of such an attempt, if successful, can hardly be doubted. By ren- 
dering more accessible a subject of great interest and importance, the study 
of English mathematicians may be recalled to a theory which, although it 
originated in England, has not received the attention it deserves, and which 
it has met with in foreign countries. 
The paper which I have the honour to submit to the Royal Society, contains 
a complete determination of the variable elements of the elliptic orbit of a dis- 
turbed planet, deduced from three differential equations that follow readily 
from the mechanical conditions of the problem. In applying these equations, 
the procedure is the same whether a planet is urged by the sole action of the 
central force of the sun, or is besides disturbed by the attraction of other 
bodies revolving about that luminary ; the only difference being that, in the 
first case, the elements of the orbit are all constant, whereas in the other case 
they are all variable. The success of the method here followed is derived from 
a new differential equation between the time and the area described by the 
planet in its momentary plane, which greatly shortens the investigation by 
making it unnecessary to consider the projection of the orbit. But the solution 
in this paper, although no reference is made to the analytical formulas of the 
theory of the variation of the arbitrary constants, is no less an application of 
that method, and an example of its utility and of the necessity of employing it 
in very complicated problems. 
1. If S represent the sun and P, P' two planets circulating round that lumi- 
nary, it is proposed to investigate the effect of the attraction of P' to disturb 
the motion of P and to change the elements of its orbit. We here confine our 
attention to one disturbing planet ; for there is no difficulty in extending to 
any number, the conclusions that shall be established in the case of one. 
The positions of the planets P and P' may be ascertained as usual by the 
rectangular coordinates x, y, z and x', y',z ' ; x,y,x',y being contained in a 
