ON THE THEOIIY OF POINT-GROUPS. 75 



short, but bears the unmistakable impress of that prolific genius which, 

 upon the suggestion offered by any particular theorem, in no matter what 

 branch of pure mathematics, at once sought its appropriate generalisa- 

 tion. In Chasles's ' Apercju historique ' (published in 1839) Cayley had 

 come across that demonstration of Pascal's theorem which we have seen 

 already employed by both Gergonne and Pliicker. The demonstration of 

 the property of cubics involved is, he says, ' one of extreme simplicity. 

 Let U=0, V=0 be the equations of two curves of the third order, the 

 curve of the same order which passes through eight of their points of 

 intersection (which may be considered as eight perfectly arbitrary 

 points), and a ninth arbitrary point will be perfectly determinate. Let 

 U„=;0, V„=0 be the values of U, V when the co-ordinates of this last 

 point are written in the place of x, y. Then UVo — UoV=0 satisfies the 

 above conditions, or it is the equation to the curve required ; but it is an 

 equation which is satisfied by all the nine points of intersection of the 

 two curves, i.e., any curve that passes through eight of these points of 

 intersection passes also through the ninth.' He then generalises the 

 form of equation used in the proof by forming U=M,._,„U„i-f-t?,._„V„, where 

 ■2*1— mi ^,-,1 3.re two polynomials of orders r — m, r — n with all their co- 

 efficients complete, and proceeds to consider how many arbitrary con- 

 stants are at disposal in this equation. At first sight it would appear 

 that there are ((r — m))-\-[{r —n)) — 1, this being the number of arbitrary 

 constants in ?(,_,„, Vr_„, less one removed by division (and this was the 

 erroneous conclusion arrived at by Pliicker when dealing with surfaces in 

 which r>m + n) ; but when we consider that, if r > m -f n, we may take 

 «<,._„i=w,._,„_„V,„ and v,._„ = — "^^-m-iiU,,,. and that then UeeO, we see that 

 {(r — m — n)) conditions exist among the arbitrary constants of u,_„i, v,_„ 

 (viz. those obtained by equating to zero the coefficients of ?^,._,„_„), and 

 that therefore there are only ((r — m)) + {{r — n)) — l — ((? — m — n)) inde- 

 pendent arbitrary constants at disposal. When r-^m + n — l, or m + n — 2, 

 {{r — m — «))=0, and it is therefore immaterial whether we consider these 

 cases as subject to the law affecting the cases in which r<«i-j-?t, where 

 they really belong, or under that of r>m-\-n; the simplest plan is to 

 include them under the latter and to say that when r>m-\-n — .3 the 

 degrees of freedom of a C,. through the points of intersection of the given 

 C„„ C„ are {{r—m))-\-{{r — 7i)) — 1 — {(r—m — n)) ; whereas \i r<vi-\-n—'i 

 the degrees of freedom are {{^•—m))-\-{{r — «)) — 1, which agrees with 

 Jacobi's result for r-=m-\-n — 3. 



The above statement which is suVjstantially Cayley's own, deals only 

 with the degrees of freedom of the C,. ; but the question may also be put 

 in other ways, for instance : How many conditions are imposed upon the 

 coefficients of any C,. by constraining it to pass through the vm points of 

 intersection of a given C^, and a given C„ ? And : How many equations 

 of condition must subsist among the co-ordinates of mn points on a given 

 C„ if they are the points of intersection of the C„ with a C,„ ? Since, in 

 general, a C,. has ((r)) — 1 degrees of freedom, and since we have shown 

 that, if r>TO-i-TO— 3, a C,. under the given conditions has {{r — n)) -f- {{r—m)) 

 — 1 — {{r — m—n)) degrees of freedom, it follows that the number of con- 

 ditions imposed by the mn points must be the difference between these 

 numbers, i.e., exactly mn ; but if r<m-\-n — 3, the degrees of freedom of 

 the C^ were found to be {{r—m)) — {{r—n)) — \, and therefore the number 

 of conditions imposed by the mn points on the constants of the C,. is, in 

 that case, {{r)) — \—{{r—n)) — {(^r—m)) + \=mn—{{r -m — n)). Again, 



