ON THE PRESENT STATE OF THE THEORY OF POINT-GROUPS. 125 



each variable appears. The latter is the more difficult pi-oblem, and 

 admits of complications in which the interpretation of certain equations 

 as correspondences is of great value (see infra, p. 129). 



In this paper he finds by induction, without a rigorous proof, a 

 formula for the number of solutions of a system of equations in k inde- 

 pendent variables (each equation being symmetrical in all the variables), 

 the system consisting of a number of equations equivalent to i + \ inde- 

 pendent equations ; so that k—i—\ of the variables must have arbitrarily 

 assigned values before the expression ' number of solutions ' can have a 

 meaning. When k—i—\ have been so chosen, the 'number of solutions' 

 means the number of different ways in which the remaining i + \ L,an be 

 found to satisfy any i + \ equations of the system. The number (see 

 formula (A), infra) is made to depend on the sums and differences of the 

 numbers of common solutions of pairs of systems of equations (in square 

 brackets in (A) below), one system of each pair being of the same kind 

 as the original system, but equivalent to fewer than i + l independent 

 equations ; while the second system of the pair is either precisely one 

 equation, symmetrical in all the variables, or consists of a system equiva- 

 lent to 2, or 3, . . ., or i + l independent equations, involving only k—\, 

 k — 2, . k—i variables respectively. As an important example of a 

 system of equations of the assumed nature, he considers the original 

 system to consist of all the equations formed by equating to zero every 

 ^-rowed determinant of the following matrix of k + i columns and 

 k rows, 



01 (^l). 02 (^l), • • • 0t + «(^l)j 



'1 ('^2)1 02 ('^2). • • • 0A + ;, (^2): 



01 {\), 02 i^k). 



'Pk.i{K) 



where ^1, . . . (^4+,; are integral functions of the mth degree of the 

 single variables enclosed in the brackets, these variables A,, . . . /\. being 

 all independent. Such a matrix is more shortly written as \\k + i\\,., and 

 the number of solutions as {k + i),,. This notation is also employed when 

 the ^/s are functions of more than one variable each, the variables beino' 

 then connected by k relations (see infra, p. 127). The number of common 

 solutions of all the A'-rowed determinants it contains is known to be equal 

 to the number of common solutions of the i + l determinants, 



0i(^i)j • • • 0A-i(''^i). 0((^i)| 



0;,-i(^2), 0((^2) 



01 (\.). • • • 0;.-i('\), h{\)\ 



. k-\-i in turn, j^rovided the A — 1 -rowed deter- 

 •Pi ('"^i). • • • 0i-i(^i) 



where <=A-, A; + l, . . 

 minants of the matrix 



do nob all vanish. 



01 OV), 



0*-i(\.) 



