300 
PROFESSOR DONKIN ON THE 
equations of the planetary theory. This investigation, if interesting at all, will pro- 
bably be so to the mathematician rather than to the astronomer. I think, however, 
that if the theories of physical astronomy were more frequently treated rigorously 
and symmetrically, apart from any approximate integrations; and if, when the latter 
are introduced, more care were taken to give a clear and exact view of the nature of 
the reasoning employed, it might be possible to draw the attention and secure the 
cooperation of a class of mathematicians who now may well be excused, if, after a 
slight trial, they turn from the subject in disgust, and prefer to expatiate in those 
beautiful fields of speculation which are offered to them by other branches of modern 
geometry and analysis. 
The contents of the two last Sections are more or less closely connected with the 
subjects of various memoirs by other writers, especially Professor Hansen and the 
Rev. B. Bronwin. I cannot pretend to that degree of acquaintance with them which 
would enable me to give an exact statement of the amount of novelty to be found 
in my own researches. I believe it is enough to justify me in offering them to the 
Society ; beyond this I make no claim. 
Oxford, Feb. 15, 1855. 
Section IV. 
49. The following theorems were demonstrated in the former part of this essay, 
and are recapitulated here for convenience of reference. (As before, total differen- 
tiation with respect to the independent variable t will, in general, be denoted by 
accents, which will be used for no other purpose.) 
Theorem I. — If X be a function of n variables j?,, a? 2 , and if 3 /,, 3 / 2 , be n 
other variables connected with the former by the n equations 
dX dX dX 
dx^ dx^ y^'‘"'dx^ (^b.) 
then will the values of x^, x^,....x^, expressed by means of these equations in terms 
of 3 / 1 , .... 3 /„, be of the form 
dY dY dY , , 
X,-dy,X^-dy^,...^n-dyf • ( 51 .) 
and p be any other quantity explicitly contained in X, then also 
^-^+^=0 
dp ' dp 
(52.) 
(the differentiation with respect to p being in each case performed only so far as 3 ? 
appears explicitly in the function). 
The value of Y is given by the equation 
^=-0^)-\-{x,)y,-\-{x^)y^-\-...-\-{.i\)y„, (53.) 
where the brackets indicate that .r, ... are supposed to be expressed in terms of 
... 3 /„ (arts. 2, 3.). 
