6:2 Mr. A. Caylcy on a Theorem relating to Surfaces, 



before-mentioned branch is this conic; that is, the conic APA 

 by its intersection with an arbitrary plane traces out a conic ; 

 or, what is the same thing, the sheet, the locus of the conic 

 APA', is met by an arbitrary plane in a conic, that is, the sheet 

 is a surface of the second order ; and the given surface thus 

 includes a surface of the second order, and is therefore made up 

 of two surfaces of the orders m and % respectively. The demon- 

 stration seems to me to add at least something to the evidence 

 of the theorem asserted, but I should be glad if a more simple 

 one could be found. Analytically, the theorem is — " If 



{at,y,z, az + fBy + yz) m+n , 



where (a, ft, 7) are arbitrary, break up into factors (a?, y f z) m , 

 (x, y, z) n , rational in regard to (x, y, z), then {on, y, z, w) m+n 

 breaks up into factors (oc, y, z, w) m , (x, y, z, w) n , rational in 

 regard to (x, y, z, w)" It would at first sight appear that 

 {"> A 7) being arbitrary, these quantities can only enter into 

 the factors of (x, y, z, ax + fBy + yz) m+n through the quantity 

 ax i-/3y + yz; that is, that the factors in question, considered as 

 functions of (x, y, z, «, /3, 7), are of the form 



(x, y, z, ax + /3y + yz) m , (x, y, z, ax + j3y + yz) n ; 



and then replacing the arbitrary quantity ax + f3y + yz hyw, the 

 factors of (x, y, z } w) m+n will be (x, y, z, w) m , (x, y, z, w) n . But 

 this reasoning proves too much ; for in a similar way it would 

 follow that if (x } y, ax -f fty]" l+n , where a, /3 are arbitrary, breaks 

 up into the factors (x, y) m , (x, y) n , rational in regard to (x, y) 

 (and qua homogeneous function of two variables it always does so 

 break up), then (x, y, z) m+n would in like manner break up into 

 the factors (x, ?/, z) m , (x, y, z) n , rational in regard to [x, y, z). 

 And a simple example will show that it is not true that the factors 

 of (x, y, ux-\-{3y) m+n only contain (a, j3) through ax + {3y; in 

 fact, if the function be = x°~ -f y l + {ax + /3y) 2 , then the factor is 



which cannot be exhibited as a function of a, /3, ax-\-j3y. lam 

 not acquainted with any analytical demonstration; the geome- 

 trical one cannot easily be exhibited in an analytical form. 



2 Stone Buildings, W.C., 

 November 26, 1862. 



