ON THE PRESENT STATE OF THE THEORY OF POINT-GROUPS. 127 



had given 1 an extension of this principle to curves of any deficiency 

 «,2 without, however, formally proving it, and it is at this stage that 

 Brill took up the subject. He gives an algebraic proof of Cayley's formula 

 for the number of united points of one correspondence on a given curve of 

 deficiency j), and he finds, moreover, the proper extension to curves 

 of any deficiency of the well-known algebraic theorem : ' if betiveen the 

 points X, y of a straight line, there exists a relation ,p{^, y)=0 by means of 

 which u points x correspond to a j)oint y and X 2yoints j to a point x ; and 

 further, if by means of a second relation (p'(x, y)=0 k' points x correspond 

 to \' points y ; then the number of jmirs of jjoints lohich satisfy both 

 relations is {<P<P')=kX' + >:'\.' The first relation ^(a;, 2/)= is said to establish 

 a correspondence (k, X) between the points on the straight line ; the 

 second, <p'{x, y)=0, a correspondence (ic', X'), and (ff) gives the number of 

 pairs of points which satisfy both correspondences. 



Brill's extension is as follows -.—Given a fixed point z on a curve t oj 

 deficiency p, and two movable points, x, y, on the same curve, and let the 

 two relations, <l> (x, y,z)=0, (^'{k, y,z)=0 hold, which, regarded as functions 

 ofx have k, k' points of intersection, respectively, loith i(x)=0, of which 

 /3, /3', respectively, coincide with the point z, and y, y' with the point y ; ^ 

 and which, regarded as functions of j, have 1, V p)oints of intersection, 

 respectively, with f (y)=0, of ivhich a, a', respectively, coincide with _ the 

 point z, and y, y' with the jyoint x ; then the number of pairs of points, 

 x, y, {each point being distinct from the other and not coinciding with z), 

 ivhich satisfy both relations is given by ((p(l)')=KX' + K'X—2pyy', 



^^^^^ [x=l-y-a,X'=l'-y-a'. 



The first application of this formula is to find in hov) many ivays three 

 points on a curve f=:0 can be chosen, so that a singly i^ifinite system of 

 curves, selected from a given triply-infinite system, may pass through Jhein. 

 This is a simple case of the problem already referred to on p. l'J5 (viz. 

 ;5._3^ 1=1), but now we have to deal with a base-curve of any deficiency 

 p, instead of the straight line, and thus it is impossible to eliminate y and 

 to obtain equations in three independent variables ^i, x^, ^3- We obtain 

 a matrix similar, but not identical, to that on p. 125, viz. : — 



?>i(^i2/i)> V>2(^i2/i). <}'s(^iyi)> 1>i{^\y\)\ 



fii^-ii/ih 02(^22/2)) fA^iV-i)^ f4{^2y2) 

 <pi{^3ys)> ni^zVz)' <p-i{^zyz\ <Pi{^zyz). 



and further, the three e(\\xa.i\onsf {x^y^) = 0, f {x.^'>)=0, f {x-^y^)=Q. 



As before remarked, however, the formula (A) still holds and gives us 

 (3-|-l)3=[(3)3(.3).,]— (3 — 1)3 but for this simple case it is worth while to 

 work out the problem directly in the first place, without using formula 

 (A), as it afi"ords insight into the geometrical meaning of a correspondence 



Observe, first, that before a finite number of solutions can be found, 

 one point, z, on /=0, must be assumed arbitrarily, since k-i-l = l. 



' Comptcs Eendus, vol. Ixii., 1866, p. 586. 



- Cayley's result is that the number of ' united points ' is to + w + 2/;Z', where ^ is a 

 quantity afterwards known as the Wertigkeit of the correspondence curve. 

 3 j3 is said to be the Wertlglteit of (^ at .r = z, 7 at a; = y ; 



3' „ „ „ „ <^'at3: = 2^,y ata> = y; 



a. „ „ „ „ <P at 2/ = 2, 



a' „ „ „ ,, ^'aty = c. 



