ON THE THEORY OF TOINT-G ROUPS. 67 



the priority in publication which is undoubtedly his.^ This, however is 

 a very small matter : Gergonne's contribution to the elucidation of pro- 

 blems connected with the intersections of curves is insignificant compared 

 with Pliicker's. It was Pliicker who derived from Lame's equation of a 

 system of curves the theorem -which threw fresh light upon the so-called 

 Cramer Paradox, which had baffled mathematicians for more than a hun- 

 dred years. And it was Pliicker who, simultaneously with Jacobi (1804- 

 1851), first ventured upon a line of research which afterwards proved a 

 fruitful source of theorems in the theory of point-groups — the investiga- 

 tion, namely, of the conditions which must exist among the co-ordinates 

 of certain points if they are known to be the points of intersection of two 

 curves of given (differing) orders. 



The problem which first led Pliicker to consider the paradox was that 

 of determining the highest degree of osculation possible between a curve 

 of order n and one of order m. This question is treated in a footnote to 

 an earlier chapter - of the work just mentioned, and its solution is made 

 to depend upon the establishment of a new theorem, viz. that all curves 

 of the nth order, which pass through^ {(n)) — 2 given points intersect each 

 other also in the samen'^—{(n)) + 2=((n-3)) points. In this passage the 

 paradox is not explicitly mentioned ; but in a paper published in the 

 same year in Gergonne's Annales "^ Pliicker speaks of it, describing it as 

 the fact that in certain cases two curves of the same order may cut each 

 other in at least as many points as are required to completely determine 

 one of them. 'Cramer,' he continues, 'dans son "Introduction a 

 I'analyse des courbes algebriques," est le premier, je crois, qui ait signale 

 cette espece de paradoxe qui s'explique aisement en remarquant que, 

 lorsqu'il est question du nombre des points necessaires et suffisants sur un 

 plan, pour determiner compl^tement une courbe d'un degre determine, on 

 sous-entend toujours que ces points sont pris au hasard, et ne sont lies 

 entre eux par aucune relation particuliere.' He establishes his new 

 theorem in almost the same words here as in the other passage ; the 

 application is to the theory of the conjugate points of conies. 



The second volume of the ' Analytisch-geometrische Entwicklun^en ' 

 was published in 1832 : in this Pliicker returned to the subject of^'the 

 paradox,'' and remarked that Cramer had indicated the analytical explana- 

 tion, viz. that the n- linear equations which correspond to the w- points 

 of intersection of two curves of order to must, if n>3, be such that one 

 or more, arbitrarily chosen from among them, are conditioned by those 

 which remain ; he adds that a geometrical interpretation of this explana- 

 tion is needed. His own new theorem affords this geometrical interpreta- 

 tion, and he therefore reproduces it once more, with a proof which, when 

 slightly elaborated, is substantially as follows : — 



Assume ((«)) — 2 arbitrary points in the plane, take any two curves of 

 order 7i through them, 17=0, V=0, which, in general, are completely 

 determined if we know one more point, not the same, on each. Suppose 



' See Kotter, ' Die Entwickelung der synthetischen Geometric von Monge bi.s 

 auf Staudt,' Jahresber. d devtscJt. Math.-Verein., vol. v. (1901), p. 226. Clebsch 

 (loc. cit., p. 19) ascribes the priority to Pliicker, without mentioning Gereonne 



= P. 228. 



' ((»)) is written throughout for ^ («. -t- 1) (» + 2). 



* 'Recherches sur les courbes algebriques de tous les degres,' Gerg. Ann. 

 vol. xix. (1828), pp. 97-106 ; also Works (Leipzig, 1895) pp. 7C-82. 



' P. 242. 



