74 REPORT — 1903. 



left-hand side of ^;=0 multiplied in its turn by an arbitrary function of 

 degree (m — 3), and there are thus {(n — 3)) + ((j7z — 3)) arbitrary constants 

 (homogeneous) involved. It is thus seen that ({m — 3)) + ((n — 3)) of 

 equations (A) follow from the rest, and the number of independent 

 equations is therefore {(vi + n — 3)) — ((m — 3)) — {(n — S))=^mn — 1. And 



since there are vi7i quantities ^ , they can be solved from these nin — 1 



IXk 



equations, and the results in terms of the simultaneous roots can be 

 substituted in the ((m — 3)) + {{n — 3)) remaining equations, which are 

 therefore the equations of condition among the 2m7i simultaneous values 

 of ,T and 2/. But since it is possible that((??i — 3)) + ((w — 3)) may be >^2mn, it 

 is clear, says Jacobi, that this number of equations of condition is too many ; 

 and he then proceeds to show that ((«i — n — 3))of these must follow from the 

 rest, and that, therefore, the real number of independent equations of condi- 

 tion is, as he found before, ((wi — 3)) + ({n — 3)) — {{m — n — 3))=mn~3ii + l. 

 Into- this part of his discussion it is not worth while to enter here, as, 

 once more, no criterion is established of the independence of the equations 

 of condition when found in this manner. 



The problem of determining the number of arbitrary constants at 

 disposal [i.e., non- homogeneous) in an equation of given order r, which is 

 satisfied by the simultaneous roots of two other given equations of differing 

 orders m, n — or, as it may be more shortly expressed, the problem of deter- 

 mining the degrees of freedom of a C,. — had been solved by Bezout (for 

 any number of variables) as early as 1774 ; but the ' Theorie des equa- 

 tions algebriques,' which is devoted to the genei'al problem of the elimi- 

 nation of variables from a system of simultaneous equations, appears to 

 have fallen temporarily into oblivion, and is not referred to by any writer 

 of this period. As far as the equations of curves are concerned, Pliicker 

 had only dealt with the case in which r=m^=n, in which there is pre- 

 cisely one degree of freedom, as is at once appai'ent from the form of the 

 equation U + AV=0 ; for even the theorem which dealt with the inter- 

 sections of a C,. and a C,,, is based upon the discussion of a C,. through the 

 points of intersection of two other C,s, one of which is degenerate. In 

 dealing with surfaces Pliicker had come across the more general case, but 

 he gave it at first a wrong solution. Jacobi, besides the more obvious 

 case of r=^m = n, had treated, as we have just seen, the case for curves in 

 which r^m + n — 3, and had shown that the degrees of freedom are then 

 ((?n — 3)) -)-((« — 3)) — 1. In the case of surfaces he also found the correct 

 number, although the explicit problem before him there — as also before 

 Pliicker — was that of the number of equations of condition which 

 must hold among the common points of three surfaces of degree 

 m, n, r, and only intermediately that of determining the degrees of 

 freedom of a surface through the curve of intersection of two others. 

 "When curves are concerned, the problem of determining the degrees of 

 freedom of a curve through the intersections of two other curves and the 

 problem of determining the number of equations of condition which must 

 subsist among the co-ordinates of certain points in order that they may 

 be the points of intersection of two curves of given orders are directly 

 connected with one another, as will appear from an account of two papers 

 by Cayley (1821-1895), which fall within the period under discussion. 



The first of these, entitled ' On the Intersections of Curves,' appeared 

 in 1843,' when its author was only twenty-two years of age. It is 



' Camlridge Mathematical Journal, vol. iii. pp. 211-213 ; Worlts, vol. i. pp. 25-27. 



