Mr. J. J. Sylvester on a ?iew Class of Theorems. 367 



as our compound characteristic (a function, it will be observed, 

 of five letters, <r, y, s, t 9 u) ; if its determinant have two equal 

 roots, L has two consecutive points in common with the inter- 

 section of P and Q, i. e. passes through a tangent to that in- 

 tersection ; if it have three equal roots, L has three consecu- 

 tive points in common with the said intersection, i. e. is an 

 osculating plane thereto ; if its fifteen first minors admit of all 

 being made simultaneously zero, L has a double contact with 

 the intersection of P and Q, i. e. it is a tangent plane to some 

 one of the four cones of the second order containing this in- 

 tersection; if the same linear function of fi which enters into 

 all these first minors be contained cubically in the complete 

 determinant, then the plane L passes through four consecutive 

 points of the intersection of P and Q, and the points where it 

 meets the curve will be points of contrary plane flexure ; and, 

 as it seems to me, at such points the tangential direction of 

 the curve must point to the summit of one or the other of the 

 four cones above alluded to*. In assigning the conditions for L 

 being a double tangent plane to the intersection of P and Q, 

 we may take any three independent minors at pleasure equal 

 to zero. One of these may be selected so as to be clear of 

 the coefficients of L ; in fact, the determinant of P + /u,Q will 

 be a first minor of P + yitQ + Lw; fju may thus be determined 

 by a biquadratic equation ; and then, by properly selecting 

 the two other minors, we may obtain two equations in which 

 only the first powers of the coefficients of #, y, z, t in L appear, 

 and may consequently obtain L under the form of 



(ae + a)x + [be + $)y + (ce + y)z + (de + h)t, 



where a, a; b, /3; c, y; d, 8 will be known functions of any 

 one of the four values of fju. The point of contact being given 

 will then serve to determine e, and we shall thus have the equa- 

 tion to each of the four double tangent planes at any given 

 point fully determined. 



In the foregoing discussions I have freely employed the 

 word characteristic without previously defining its meaning, 

 trusting to that being apparent from the mode of its use. It 

 is a term of exceeding value for its significance and brevity. 

 The characteristic of a geometrical figure f is the function which, 



* If this be so, then we have the following geometrical theorem : — " The 

 summit of one of the four cones of the second degree which contain the inter- 

 sections oftiuo surfaces of the second order drawn in any manner respectively 

 through two given conies lying in the same plane, and having with one another 

 a contact of the third degree, will always be found in the same right line, 

 namely in the tangent line to the two given conies at the point of contact." 



f More generally, the characteristic of any fact or existence is the func- 



