70 REPORT— 1903. 



determined. If the rest, which are nq—{{q — 3)) in number, satisfy an equa- 

 tion Cg = of order q, then, since Aj,C,^ = is one of the equations of order 

 n satisfied by the ((w)) — 2 = ((|>)) — 1 -{-nq — ((5' — 3)) couples of values, it 

 follows from the first algebraical statement of the original theorem that 

 it will also be satisfied by ((?i — 3))=(((7 — 3)) + 7?;; — ((p)) + 1 other couples 

 of values ■. but since every equation of order n has nq couples of values which 

 are common to it and to C,^ =: 0, it follows that the nq — {i^q — 3)) above- 

 mentioned couples of values which satisfy C,j = must lead to ((g — 3)) 

 others, which also satisfy this equation of order q. 



Pliicker's final utterance on the intersections of plane curves occurs in 

 the ' Introductory Considerations,' which form the first chapter of his 

 * Theorie der algebraischen Curven,' published in 1839. He there rejaeats 

 the geometrical formulation of the original theorem, and also formulates the 

 new theorem, geometrically, thus : All curves of the nth order ivhich jjass 

 through nq — ((q — 3)) points arbitrarily assumed on a given curve of order 

 q cut this curve again in ((q — 3)) more fixed points. He further considers 

 what possibilities exist for the distribution of the arbitrary points on two 

 fixed curves of orders p and q respectively, where pi + q=n, in order that all 

 curves of order n through these arbitrary points may intersect each given 

 curve again in a certain number of fixed points. These considerations 

 lead him to state : If of the n- points of intersection of two curves of order n, 

 nq — ((q — 3)) lie on a curve of order q, then a curve of order n — q 

 passes thro^igh the remaining n{n — (\) 2)oints. This, as he points out, is 

 an improvement on Gergonne's theorem, inasmuch as it obtains the same 

 result with a smaller number of assigned points. The closing paragraph 

 of this chapter is devoted to historic considerations. In it Pliicker refers 

 to the passages in his former book, and to his papers in Gergonne's 

 Annales, and once more draws attention to Cramer as the originator of 

 the paradox.^ He then goes on to explain that his paper in Crelle's 

 Journal, although published in the sixteenth volume, was in the editor's 

 hands at the same time as one of Jacobi's which appeared in the fifteenth 

 volume. He adds that his own had been intended for the first volume of 

 Liouville's Journal (which replaced Gergonne's Annales at about this date) 

 and that ' a celebrated analyst had occasioned its preparation by a verbal 

 observation about the difficulty of extending the relations which connect 

 the roots of an equation in one variable to the case of the simultaneous 

 roots of a system of two or more equations among two or more variables. 

 . . . This is why it was written in French, and clothed in algebraic form.' 



It was characteristic of Pliicker's genius that he consciously limited 

 the scope of his mathematical investigations to one particular domain- — ■ 

 that of analytical geometry — v/ithin which, indeed, he found ample room 

 for the employment of his rich imagination. This probably accounts for 

 the fact that his writings on the intersections of curves are completely 

 uninfluenced by the theory of functions, although his lifetime precisely 

 covei's the years in which this new branch of pure mathematics was being 

 created. The account of the influence of the theory of functions on the 

 theory of higher plane curves will fall into a later division of this RejDort, 

 but it may be mentioned here that the interest which attaches itself to 

 the paper of Jacobi's referred to by Pliicker is partly due to its close 



' For this point, cf. Part 11. of this Feport, § 6, last paragraph, Brit. Assoc. 

 Iiej)ort, 1902. 



