366 Mr. J. J. Sylvester on a new Class of Theorems, 



Now I have established the following law : — 



The whole of a system of rth minors being zero, implies 



only (r + 1) 2 equations, i. e. by making (r + l) 2 of these minors 



zero, all will become zero ; and this is true, no matter what 



may be the dimensions or form of the complete determinant. 



But furthermore, if the complete determinant be formed from 



a quadratic function, so as to be symmetrical about one of its 



ff4-l)(r + 2) 

 diagonals, then ~ - only of the rth minors being 



zero, will serve to imply that all these minors are zero. Of 



course, in applying these theorems, care must be taken that 



}f _i_ i )(r 4- 2) 

 the (r+1) 2 or * ~ - ; selected equations must be mutu- 



z 



ally non-implicative, and shall constitute independent condi- 

 tions. 



In the application I am about to make of these principles, 

 we shall have only to deal with a system ofjlrst minors and of 

 a symmetrical determinant. If three of these properly selected 

 be zero, from the foregoing it appears that all must be zero. 



Now let U and V be characteristics of two conies, i. e. let 

 each be a function of only three letters, it may be shown (see 

 my paper in the Cambridge and Dublin Mathematical Journal 

 for November 1850) that the different species of contacts be- 

 tween these two conies will correspond to peculiar properties 

 of the compound characteristic XJ + fiV. 



If the determinant of this function have two equal roots, the 

 conies simply touch ; if it have three equal roots, the conies 

 have a single contact of a higher order, i. e. the same curva- 

 ture ; if its six first minors all become zero simultaneously for 

 the same value of /jl, the conies have a double contact. If the 

 same value of /ul, which makes all these first minors zero, be 

 at the same time not merely a double root (as of analytical 

 necessity it always must be), but a treble root of 



|— |(U + ,*V)=0, 



then the conies have a single contact of the highest possible 

 order short of absolute coincidence, i. e. they meet in four 

 consecutive points. 



The parallelism between this theory and that of two qua- 

 dratic functions P, Q,and one linear function L # of four letters, 

 say x 9 y 9 % 9 t, is exact f. For let P+Lw + ^Q be now taken 



* Observe that P=0, Q=0, L=0 now express the equations to two co- 

 noids and a plane respectively. 



t This parallelism may be easily shown analytically to imply, and be 

 implied, in the geometrical fact, that the contact of the plane L with the 

 intersection of the two surfaces P and Q, is of exactly the same kind as the 

 contact (which must exist) between the two conies which are the intersec- 

 tions of P and Q respectively with the plane L. 



