364? Mr. J. J. Sylvester on a new Class of Theorems. 



suppose that the determinant in respect of those letters of 

 U + z^V contains i. pairs of equal linear factors of //, ; then it is 

 possible, by means of i linear equations instituted between the 

 letters, to make U and V each become functions of the same 

 ?n — 2i orders; and conversely, if by i equations between the 

 letters U and V may be made functions of the same m— c 2l 

 orders, the determinant of U + /aV considered as a function of 

 fju will contain i quadratic factors. 



Thus when m=z2n and i=n, U and V will each become 

 functions of zero orders, i. e. will both disappear, provided 

 that on the institution of a certain system of n linear equations, 

 among the letters of which U and V are functions, the deter- 

 minant of (U + yuV) is a perfect square,— which is the theorem 

 given in the article referred to. 



So [ex. gr.) if U and V be quadratic functions of four letters, 

 and therefore the characteristics of two conoids, j | (U + yu-V) 

 being a perfect square, expresses that these conoids have a 

 straight line in common lying upon each of their surfaces. 



If U and V be quadratic functions of three letters only, and 

 admit therefore of being considered as the characteristics of 

 two conies, |~ ~| (U-f/x-V) containing a square factor, is indi- 

 cative of these conies having a common tangent at a common 

 point, i. e. of their touching each other at some point; for it 

 is easily shown that the disappearance of two orders from any 

 quadratic function by virtue of one linear function of its letters 

 being zero, indicates that the line, plane, &c. of which the 

 linear function is the characteristic is a tangent to the curve, 

 surface, &c. of which the quadratic function is the character- 

 istic. 



I pass now to a generalization of the theorem which shows 

 how to express, under the form of a double determinant, the 

 resultant of one linear and two quadratic homogeneous func- 

 tions of three letters (which I should have given in the original 

 paper, had I not there been more intent upon developing an 

 ascending scale than of expatiating upon a superficial ramifi- 

 cation of analogies), and which constitutes my Second addition 

 to that paper, to wit — 



If U and V be homogeneous quadratic, and L x L 2 . . . ,L n 

 homogeneous linear functions of («-f 2) letters a?j #%. . . . x n+2 , 

 the determinant of the entire system of n + 2 functions is equal 

 to 



]{\V+/*V + 1 L l t l +Lj 9 +...+Lj n }; 



X, /j, x lf <r 2 , ...x n +2 t x t 2 ... tn 



the demonstration is precisely similar to the analytical one 



