ON THE PKESENT STATE OF THE THEORY OF POINT-GROUPS. 129 



of a singly- infinite system. The number of such pairs is obviously the 

 combinations in twos of the M — 1 points besides z, which lie on/=0, i.e. 



it is '^ -. If, therefore, we write k^=k'^=\=z\'=M. — '2 and 



■y=7'=l in the formula given on p. 127 for (i^^') (dividing it, however, 

 by 2, since our correspondences are symmetrical), and then subtract 



'- from this, we obtain the number of triplets, viz. (M — 2)^ — 



p - ^(M-l)(M-2)=i(M-2)(M - ?,)-p. 



Now compare the steps of this process with the formula (A) for this 

 case, i.e. with (3 + l)3 = [(3);j(3)3] — (3 — 1)3 and we see that the two cor- 

 respondences employed, exactly similar, were the determinants, the number 

 of whose solutions was denoted by (3)3, (3)3, while (3 - 1)3 gives the num- 

 ber of those solutions which needed to be subtracted from the total number. 



In more complicated problems, formula (A) is used at once, and the 

 evaluation of the number of common solutions of the equations in the 

 square brackets (the number and form of these equations is given above, 

 pp. 125 and 126) is performed by interpreting these as correspondence equa- 

 tions (cf. p. 124), provided we know how many points correspond to one in 

 each correspondence (they are always symmetrical, as is seen at once from 

 the form of the determinant) and also the ' Wertigkeit ' of the points 

 x—y, kc. (Here again, by symmetry \_<f\iy=[<p\:= .... &c., for the 

 determinant will vanish identically exactly the same number of times 

 whichever rows are made the same.) 



The other cases to which Brill applies the theory of correspondences 

 in the paper under consideration are : — 



{a) To find the mimher of triplets of points on a given base-curve 

 through which a doubly-infinite system of curves, contained within a ^-ply 

 infinite system, can he made to pass. 



(b) To find the number of sets of four points on a given base-curve 

 through which a triply-infinite system, of curves, contained within a &-ply 

 infinite system, can he made to pass. 



In the first of these A;=3, i=2 ; in the second A; =4, i=3. They are 

 both rather more complicated than the one we have considered in detail ; 

 the first, namely, involves finding the number of triplets of points which 

 satisfy three correspondences, for 



. (3-f2)3 = [(3 + l)3(3)3]-[(3)3(2)3] + (l)3, 



and the number of independent equations (see p. 126) involved in 

 DI3 + III3, !|3||] is 2 + 1=3, involved in [||3||3, II21I3] is l-f-2=3, and involved 

 in (1)3 is 3 ; similarly, the second involves finding the number of sets of 

 4 points which satisfy 4 correspondences, for here we have 



(4 + 3),=[(6),(4),]-[(5),(3),] + [(4),(2),]-(l)„ 



and the total number of independent equations in each term of this 

 expression is 4, namely, 3 + 1, 2 + 2, 1 + 3 and 4. Moreover, it is essential 

 to the final evaluation to take notice of the way in which the 4 inde- 

 pendent equations are grouped ; the formulae in the theory of correspond- 

 ences for finding the number of sets of 4 points which satisfy 4 equations 

 when these 4 are grouped as 3 + 1 differs from that in which they are 

 grouped as 2 + 2 ; but since the difference only appears in the terms 

 1900. K 



