,} 



Contacts of Lines and Surfaces of the Second Order. 123 



every rth minor are (r + l)th minors of the original matrix. 

 Hence it follows, that if any system of rth minor detemiinants 

 have a common factor e', the complete determinant must contain 

 at lowest the factor e^''^'^', and any system of (r— *)th minor 

 detenninants thereimto will contain at lowest the factor e'''+'^'. 



I now proceed to apply these principles to the deteimination 

 of the relative forms of conjugate quadi'atic fmictions represent- 

 ing geometrical loci of the second order. I shall begin with two 

 binaiy systems of points in a right line. 



The general characteristic U and V of two such systems may 

 be thrown under the form 



Y=ax'^ + bfJ 

 When I j (V + XU) = has its two roots equal, these sy- 

 stems have a point in common. The above forms cease to be 

 applicable, and convert into 



\J=xy 



Y=sax^ + bxy_ 

 where a?=0 represents the common point. 



Let U and V now represent two conies. When there is no 

 contact, we have as the types of their congress 

 U=a;2-|-j/2 + ^2 



V ^ ax'^ + by^ + cz^. 



The three roots of ~| (V+XW)=0 are 



\= — a \= — b \=— c, 



showing that there are three distinct pairs of lines in which the 

 intersections of U and V are contained, the equations to three 

 pairs being respectively 



(6-a)/ + (c-a)-^2 = 



{c-b)z^-\-{a-b)x'^ = Q 



{a-c)x'^+[b-c)y'^ = 0; 



the four points of the intersection being defined by the equations 

 corresponding to the proportions 



X :y : z :: \/b—c 'Vc—a '■ Va — b. 

 Now let \^^ (U -t- W) have two equal roots ; the character- 

 istics assume the form 



\]=.x'^ + y'^-\-xs 



V = ax^ + by"^ + cr^*. 



'" We may if we please make a=6 ; for it may be showa that, the equa- 



