Mr. A. Cayley on a Theorem relating to Surfaces. 61 



hexads. The hexads not belonging to one or other of the fore- 

 going classes are derived each of them from a single triad only 

 of the system, and they present themselves once among the 

 7700 hexads. This number is consequently made up as follows, 

 viz. 



280 hexads each twice = 560 

 3150 „ „ „ =6300 



840 „ „ once = 840 



4270 7700 



or there are in all 4270 distinct hexads ; and since this is less 

 than 5005, it follows that there are hexads not containing any triad 

 of the system : there must in fact be (5005—4270 = ) 735 such 

 hexads. The theorem in question is thus shown to be true. 



2 Stone Buildings, W.C., 

 November 24, 1862. 



VIII. Note on a Theorem relating to Surfaces. 

 By A. Cayley, Esq.* 



THE following apparently self-evident geometrical theorem 

 requires, I think, a proof; viz. the theorem is — "If every 

 plane section of a surface of the order m + n break up into two 

 curves of the orders m and n respectively, then the' surface 

 breaks up into two surfaces of the orders m, n respectively." 



To fix the ideas, suppose n = 2. Imagine any line meeting 

 the surface inm + 2 points, the section includes a conic which 

 meets the line in two of the m + 2 points, say the points A, A'f. 

 Suppose that the plane revolves round the line AA ; , the section 

 will always include a conic which passes through these same two 

 points A, A'; and it is to be shown that the sheet, the locus of 

 this conic, is a surface of the second order. In fact the conic 

 in question, say APA', by its intersection with an arbitrary plane 

 traces out a branch of the intersection of the given surface with 

 the arbitrary plane. And if ABA'B' be the conic in any parti- 

 cular plane through A, A', and if the arbitrary plane meet this 

 conic in the points B, B', then the branch passes through these 

 points B, B'. Imagine the plane ABA'B' revolving round 

 BB' until it coincides with the arbitrary plane; the section 

 includes a conic passing through the points B, B', and the 



* Communicated by the Author. 



f The figure referred to will be at once understood by considering A, A' 

 as the poles of an ellipsoid, or say of a sphere, ABA'B' the meridian of 

 projection, APA' any other meridian, BPB' the equator or any other great 

 circle. 



