L 71 ] 
V. On a Class of Differential Equations , including those which occur in Dynamical 
Problems. — Part I. By W. F. Donkin, M.A., F.R.S., F.R.A.S . , Savilian Pro- 
fessor of Astronomy in the University of Oxford. 
Received February 23, — Read February 23, 1854. 
THE Analytical Theory of Dynamics, as it exists at present, is due mainly to the 
labours of Lagrange, Poisson, Sir W. R. Hamilton, and Jacobi ; whose researches 
on this subject present a series of discoveries hardly paralleled, for their elegance 
and importance, in any other branch of mathematics. 
The following investigations in the same department do not pretend to make any 
important step in advance ; though I should not of course have presumed to lay 
them before the Society, if I had not hoped they might be found to possess some 
degree of novelty and interest*. 
Of previous publications with which I am acquainted, those most nearly on the same 
subject are, Sir W. R. Hamilton’s two memoirs “ On a General Method in Dynamics” 
in the Philosophical Transactions; Jacobi’s Memoir in the 17th vol. of Crelle’s 
Journal, “ Ueber die Reduction der partiellen Differential-gleichungen,” &c. ; and 
M. Bertrand’s “ Memoire sur l’integration des equations differentielles de la Meca- 
nique,” in Liouville’s Journal ( 1852 ). The relation in which the present essay stands 
to the papers just named will be apparent to those who are acquainted with them, 
and it would be useless to attempt to make it intelligible to others. 
Oxford, Feb. 21, 1854 . 
Section I. 
1. Let x u x 2 , .... x n be n variables, connected by n relations with n other variables 
3/1,3/25 ...3/„; so that each variable of either set may be considered as a function of 
the variables of the other set. Suppose then 
yi=<Pi(x 1} x 2 , .... x n ), 
[* It may be useful to specify the parts to which I should principally refer as containing what is, relatively 
to my own reading on the subject, new ; and in the present day it can hardly be required of any one to profess 
more than this kind of originality. These are — the theorem (3.), art. 1. The results of arts. 2 to 4. The 
formulae (19.), art. 7. The general form of the theorem (26.), art. 10. The processes and results of arts. 12 
to 14. The generalization, of Sir W. Hamilton’s transformation of the dynamical equations, arts. 17, 18. 
The demonstration of Poisson’s theorem, arts. 21, 22. The contents of art. 25. The method of obtaining 
elliptic elements, arts. 27 to 30. The contents of arts. 34 to 36. The solution of the problem of rotation. 
Section III.] 
