218 
MR. IVORY ON THE THEORY 
ciple of the Variation of the arbitrary constants, a method which has been much 
discussed, and which is now probably exhausted. It originated in the first 
researches on physical astronomy, and has been matured in passing through 
the hands of Euler, Lagrange, Laplace, and Poisson. The labours of these 
great geometers have raised up a general analytical theory applicable to every 
case, and requiring no more than the substitution of the particular forces under 
consideration. Invaluable as are such extensive views, the application of for¬ 
mulas constructed on considerations of so general a nature, may not always, 
be very ready or very direct, and may require much subordinate calculation. 
In important problems it may be advantageous to separate the principles of the 
method from the analytical processes with which they are conjoined, and to 
deduce the solution directly from the principles themselves by attending 
closely to the peculiar nature of the case. 
Distinguishing the two planets by their masses in and in!, the symbol R 
stands for a function of x, y, z, the coordinates of the disturbed planet in, and 
of x',y', z r , those of the disturbing planet in'. The expressions of these latter 
coordinates will be obtained by marking all the quantities in the values of 
x, y, z, with an accent, understanding that the accented quantities denote the 
same things relatively to the orbit of in!, that the unaccented quantities repre¬ 
sent in the orbit of m. The function R may be transformed in two ways, 
according as we substitute, for the coordinates, one set of values or another. It 
will be changed into a function of the four independent quantities r, v, N, i, and 
of the like four accented quantities of the planet in', by substituting the values 
of the coordinates obtained in the beginning of this section : and in this case, 
for greater precision, the partial differentials of R relatively to r and v will be 
written with parentheses, thus, and When the values of x, y, z, 
in terms of the mean motion ^ and of the six elements, a, s, e, ■&, N, i, and the 
like values of the other coordinates are substituted, R will be a function of 
the mean motions £ and and of the respective elements of the two orbits. 
In this latter transformation, the partial differentials of R will be written, 
as usual, without parentheses. It may not be improper to set down here the 
expressions of such of these partial differentials as we shall have occasion to 
refer to. 
