430 



equation of the fifth order was kindly performed for me by the Rev. 

 R. Harley. It is to be noticed that the best course is to apply the 

 method in the first instance to the forms (a, b, . .^jffl, 1) % =0, without 

 numerical coefficients (or, as they may be termed, the denumerate 

 forms), and to pass from the results so obtained to those which 

 belong to the forms (a, 6, . .^v, l) n =0, or standard forms. The 

 equation of differences, for (a /3) 2 , &c., the coefficients of which 

 are semin variants, naturally leads to the consideration of a more 

 general equation for (a /3) 2 (x -yy) 2 (# y) 2 , &c., the coefficients 

 of which are covariants ; and in fact, when, as for equations of the 

 orders two, three, and four, all the covariants are known, such co- 

 variant equation can be at once formed from the equation of dif- 

 ferences ; for equations of the fifth order, however, where the cova- 

 riants are not calculated beyond a certain degree, only a few of the 

 coefficients of the covariant equation are given. At the conclusion 

 of the memoir, I show how the equation of differences for an equation 

 of the order n can be obtained by the elimination of a single quan- 

 tity from two equations each of the order n 1; and by applying to 

 these two equations the simplification which I have made in Bezout's 

 abridged method of elimination, I exhibit the equation of differences 

 for the given equation of the order n, in a compendious form by 

 means of a determinant ; the method just employed is, however, that 

 which is best adapted for the actual development of the equation of 

 differences for the equation of a given order. 



III. " On the Theory of Elliptic Motion." By ARTHUR CAYLEY, 

 Esq., F.R.S. Received March 9, 1860. 



The present Note is intended to give an account of the results which, 

 by means of a grant from the Donation Fund of the Royal Society, 

 I have procured to be calculated for me by Messrs. Greedy and Davis, 

 and which are contained in a memoir presented to the Royal Astro- 

 nomical Society, entitled " Tables of the Developments of Functions 

 in the Theory of Elliptic Motion." The notation employed is 



r y the radius vector ; 



/, the true anomaly ; 



a, the mean distance ; 



e, the excentricity ; 



ff, the mean anomaly ; 



