1867.] 



which satisfy given conditions. 



463 



V. " Second Memoir on the Curves which satisfy given conditions; 

 the Principle of Correspondence/' By Professor Cayley, F.R.S. 

 Received April 18, 1867. 



(Abstract.) 



In the present memoir I reproduce with additional developments the 

 theory established in my paper " On the Correspondence of tvfo points on 

 a Curve" (London Math. Society, No. VII., April 1866); and I en- 

 deavour to apply it to the determination of the number of the conies which 

 satisfy given conditions ; these are conditions of contact with a given 

 curve, or they may include arbitrary conditions Z, 2Z, &c. If, for a mo- 

 ment, we consider the more general question where the Principle is to be 

 applied to finding the number of the curves C of the order r, which satisfy 

 given conditions of contact with a given curve, there are here two kinds 

 of special solutions ; viz., we may have proper curves C** touching (spe- 

 cially) the given curve at a cusp or cusps thereof, and we may have im- 

 proper curves, that is, curves which break up into two or more curves of 

 inferior orders. In the case where the curves C*" are lines, there is only 

 the first kind of special solution, where the sought for lines touch at a 

 cusp or cusps. But in the case to which the memoir chiefly relates, 

 where the curves C are conies, we have the two kinds of special solutions, 

 viz., proper conies touching at a cusp or cusps, and conies which are line- 

 pairs or point-pairs. In the application of the Principle to determining 

 the number of the conies which satisfy any given conditions, I introduce 

 into the equation a term called the " Supplement " (denoted by the abbre- 

 viation "Supp."), to include the special solutions of both kinds. The 

 expression of the Supplement should in every case be furnished by the 

 theory ; and this being known, we should then have an equation leading to 

 the number of the conies which properly satisfy the prescribed conditions ; 

 but in thus finding the expression of the Supplements, there are difficulties 

 which I am unable to overcome ; and I have contented myself with the 

 reverse course, viz., knowing in each case the number of the proper solu- 

 tions, I use these results to determine a posteriori in each case the expres- 

 sion of the Supplement ; the expression so obtained can in some cases be 

 accounted for readily enough, and the knowledge of the whole series of 

 them will be a convenient basis for ulterior investigations. 



May 9, 1867. 

 Lieut. -General SABINE, President, in the Chair. 



Pursuant to notice given at the last Meeting, Mr. Webster proposed, and 

 Mr. Heywood seconded, the Right Hon. Sir William Bovill, Lord Chief 

 Justice of the Common Pleas, for election and immediate ballot. 



The ballot having been taken, the Lord Chief Justice Bovill was de- 

 clared duly elected a Fellow of the Society. 



