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XV. On a Class of Differential Equations, ineluding those which occur in Dyna- 
mical Problems . — Part II. By W. F. Donkin, M.A., F.R.S., F.R.A.S., Savilian 
Professor of Astronomy in the University of Oxford. 
Received February 17, — Read March 22, 1855. 
The following paper forms the continuation and conclusion of one on the same 
subject presented to the Royal Society last year, and printed in the Philosophical 
Transactions for 1854. I have however put it, as far as possible, in such a form as 
to be independently intelligible. 
The fourth Section (the first of this Part) contains a recapitulation of some of the 
most important results of the former Part, in the form of seven theorems, here enun- 
ciated without demonstration. 
In the fifth Section the method of the variation of elements is treated under that 
aspect which belongs to it in connexion with the general subject. It is applied, by 
way of example, to deduce the expressions for the variations of the elliptic elements 
of a planet’s orbit from the results of art. 30 (Part I.), on undisturbed elliptic mo- 
tion; this example was chosen, partly because the resulting expressions are required 
in a future section, and partly for the sake of incidentally calling attention to a fal- 
lacy which has been, perhaps, often committed, and certainly seldom noticed. The 
same method, under a slightly different and possibly new point of view, is applied, as 
a second example, to the problem of the motion of a free simple pendulum, omitting 
the effect of the earth’s rotation. I believe the methods of this paper might be advan- 
tageously employed in the treatment of that general form of the problem of a free 
pendulum which has been considered by Professor Hansen in his Prize Essay. I was 
unwilling, however, to attempt what might have turned out to be merely an uncon- 
scious plagiarism, without having seen the Essay in question, of which I only suc- 
ceeded in obtaining a copy on the day of writing this preface. As I now perceive 
that the investigation would be quite independent, I hope to enter upon it at some 
future time. 
The sixth Section contains some general theorems concerning the transformation 
of systems of differential equations of the form considered in this paper, by the sub- 
stitution of new variables. The most important case consists in the transformation 
from fixed to moving axes of coordinates, in dynamical problems. Some of the 
results are, I think, interesting, and perhaps new. 
The seventh and last Section contains an application of the preceding theorems, in 
connexion with the variation of elements, to the transformation of the differential 
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