On the Contacts of Lines and Surfaces of the Second Order. 331 



changes which are more particularly the objects of j'our study. 

 But Sir John Herschel has shown that, by certain artifices^ even 

 the extreme rays may be rendered visible ; and Dr. Young, by 

 an experiment most ingenious, and to his own mind, at least, 

 most conclusive, has demonstrated the analogy of the invisible 

 with the visible rays. I feel sure, therefore, that while adducing 

 and discussing the proofs of your own theory, you will be glad 

 to take the opportunity afforded by your second edition of placing 

 Dr. Young's name in the niche which Fame has left empty. 

 Believe me to be, my dear Sir, 



Yery truly yours, 

 To Robert Hunt, Esq. J. B. Reade. 



XLIX. Note on the " Enumeration of the Contacts of Lines and 

 Surfaces of the Second Order." By J. J. Sylvester, M.A., 

 F.R.S.* 



IN the month of February, 1851, I gave in this Magazine an 

 a priori and exhaustive process, founded upon the method 

 of determinants, for determining every different kind of simple 

 or collective contact capable of happening between lines and 

 surfaces, and in general between all loci (whether intraspatial or 

 extraspatial) of the second order. The question was shown to 

 resolve itself into that of determining the number of singular 

 relations capable of existing between two quadratic homogeneous 

 functions of any given degree. My object in the paper referred 

 to was actually to calculate the geometrical and analytical cha- 

 racters of these contents and singularities for intraspatial loci, 

 i. e. loci representable by homogeneous quadratics of two, three, 

 and four variables ; but I incidentally appended a statement of 

 the number of such for loci of five, six, seven, and eight variables, 

 without, however, dwelling upon the means of representing the 

 general law. This statement is, however, affected with certain 

 inaccuracies of computation which will be presently pointed out. 

 It will be at once apparent, from an inspection of the prin- 

 ciple of my method, that it remains equally applicable {mu- 

 tatis mutandis) to the more general question of determining 

 the relative sincnlarities (in character and amount) of two 

 functions, each linear in respect of two systems of variables 

 iCj, x^ . . ,Xn; a.',, a',, . . . a'„ ; which species of functions dege- 

 nerate into quadratic forms, when the two systems of variables 

 become identical so as to coalesce into a single system. Some 

 researches of Mr. Cayley into the autometamorphic substitutions 

 of quadratic forms (meaning thereby the linear substitutions 



* Communicated by the Author. 



