76 REPORT— 1903. 



these results show that the number of equations of condition which must 

 subsist among the co-ordinates of inn points on a given C„ in order that 

 they may also lie on a given C„, are ((r — m — n)), where r is the order of 

 another curve through these inn points such that r<_m + n — 3. And 

 this agrees with the theorems of Pliicker and Jacobi. For if o'=m = n, 

 T<2r — 3, provided r>2, and ((r— ?u— w))=(( — %))=((72 — 3)) ; while if 

 r=m, m>n, r<r + n — 3, provided «>2, and once more {{r — 7n — n))r= 

 ({-n}) = {{n-3)). 



Cayley, however, did not, in this paper, express his results in terms 

 of the number of equations of condition ; the problem he was generalising 

 was geometrical, and in extending it he made the geometrical statement : 

 A curve of the rth order jMSsing through the mn points of intersection of 

 two curves of the xath and nth orders respectively, may he made to j)ass 

 through ((r)) — 1 — mn + ((r— m — n)) arbitrary points if r<m + ii — 3/ i/ 

 r be greater than this value, it may be made to pass through ((r)) — 1 — mn 

 points only. And he concludes : ' Suppose r < m + n — 3, and a curve of the 

 rth order made to pass through ((r)) — 1 — mn + ((r — m — n)) arbitrary 

 points, and mn — ((r — m — n)) of the mn points of intersection above. 

 Such a curve passes through ((r)) — 1 given jjoints, and though the 

 mn — ((r — ni — n)) are not perfectly arbitrary, there appears to be no 

 reason why the relation between the positions of these points shotild be 

 such as to jyrevent the curve from being comjjletely determined by these 

 conditions. But if this be so, then the curve must pass through the 

 remaining ((r — m — n)) jjoinfs of intersection, or ive have the theorem : 

 If a curve of the rth order (r>m or n, r<m + n — 3) ^^ass through 

 mn — ((r — m — n)) of the points of intersection of two curves of the vcith and 

 nth orders respectively, it passes through the reonaining ((r — m — n)) jjoints 

 of intersection.' 



More than forty years later (in 1886), this last theorem was challenged 

 by a writer who had been influenced by Brill and Noether's work ; the 

 account of this discussion belongs to another section. 



We have seen that Sturm in 1826 extended Desargues's theorem by 

 showing that all conies through four points cut a transversal in pairs of 

 points in involution. Since these conies have an equation of the form 

 U + XV=0, the obvious extension of the term involution is to the sets of 

 n points determined on a straight line by the curves U„ + ^-V„=0 where 

 U„,V„ are of order n. Cayley, to whom the first suggestion of an 

 extension of the term is due, went, however, much further than this in 

 his new definition. In his paper entitled ' On the Theory of Involution in 

 Geometry' published in 1847, ^ he thus defines the term : //" U,V, . . . 

 be given functions of x, y, z, . . ., homogeneous of the degrees m, n, . . ., 

 and u, v, . , . arbitrary functions of the degrees r — m, r — n, . . ., then 

 */6 = uU + vV+ . . . ., Q is a function of degree r, which is in involution 

 with XJ, V, . . . ; but, as a matter of fact the questions affecting such 

 an equation as an involution are not discussed, and he at once states 

 that the question lohich immediately arises is to find the degree oj 

 generality of 9, or the number of arbitrary constants which it con- 

 tains. It may be remarked here that the consideration of systems of 

 curves whose equations involve two independent parameters, although 

 such would come under the above general form for 6 by taking 

 G=U + XV + /iW, where U, V, W are of the same degree and involve 



' Caml. and Dvhlin Math Journ., vol. ii. pp. 52-61 ; Works, vol. i. pp. 259-266. 



