66 REPORT — 1903. 



more significant fact that any conic through the four vertices of the 

 quadrilateral cuts the transversal in a pair of points belonging to the same 

 involution. This theorem was first published by Sturm ^ (180.3-1835) in 

 1826 ; his proof is algebraical, being derived from the equations which, 

 ten years previously,^ had been shown by Lame to be the necessary conse- 

 quence of the simultaneous existence of three equations of the second 

 order. He points out that the relations he thus obtains are those which 

 establish ' cette liaison remarquable qui etait nommee par Desargues invo- 

 lution de six points.' He afterwards mentions that the two pairs of 

 opposite sides of a quadrilateral inscribed in a conic can be regarded as a 

 pair of degenerate conies, and that Desargues's theorem is thus an imme- 

 diate deduction from his own more general one ; but he makes no state- 

 ment which would lead us to suppose he saw the importance of con- 

 sidering what we now call a range of points in involution, viz. an infinite 

 number of points on a straight line, such that if any two pairs are given 

 the correspondent to a fifth point is determined by the relation called in- 

 volution which holds for any six. Nor, again, is he really interested in 

 the fact that a whole system of conies passes through the points common 

 to two conies (although, of course, he is perfectly aware that a third conic 

 through these points has an equation involving one linear parameter) ; his 

 concern is with properties of the individual conic of the system, not with 

 the system itself. And the same remark must be made about Lame, 

 although the idea of a pencil of curves is due to him ^ — that is to say, he 

 found for the first time the equation, E-|-;/iE' = 0.^ of wJiat we now call 

 a pencil of curves ; his primary interest was with the conditions which 

 must subsist among the coefficients of the equations of three curves, in 

 order that they may intersect in common points, and next, in the particu- 

 lar properties which follow for conies ; with regard to curves of higher 

 order, to which the greater interest, when looked at as a system, attaches 

 itself, he simply stated the equation. 



Gergonne (1771-1859) seems to have been the first to derive any 

 property concerning the points of intersection of curves whose equation 

 is of Lame's form, as a direct consequence of this form. In 1827 he thus 

 found ■'' that i/'p(p + q) of the (p + q)^ 2^'^'^'''''^^ ^f intersection of tivo curves 

 o/orc^er (p-hq) He on a curve of order p, the remaining q(p-f-q) points lie 

 on a curve of order q : from which he obtained the corollary : Given tioo 

 systems ofxn lines in the plane, if amonr/ the m^ points of intersection of 

 the lines of one system whh the lines of the other there are 2m which lie 

 on a conic, theri the m(m — 2) remaining points all He on a curve of order 

 (m— 2). "Writing m=.3, this is, as he points out, the theorem known as 

 Pascal's. This proof of Pascal's theorem also appears incidentally in a 

 long footnote to the last chapter of the first volume of Pliicker's ' Ana- 

 lytisch-geometrische Entwicklungen,' printed in 1828 ; the preface is dated 

 September 1827, later than the publication of Gergonne's paper, and it is 

 possible that this footnote was added at the same time ; this would give 

 the priority in discovery of this particular proof to Gergonne, as well as 



' ' M6moire sur les lignes du second ordre,' Gergonne'x Annalv-i, vul. xvii. pp. ITS- 

 198. 



2 'Sur les intersections des lignes et des surfaces,' Gerg. ^«?i., vol. vii. pp. 229-240. 

 ' Clebsch, toe. cit., p. 17. 



* In his Examen des diffcrenten vu'thndes enqjloyfes pour rcsoudre les 2>roMemes de 

 gcometrie, 1818, p. 29. See Part II. of this Report, § 8 {Brit. Assoc. Report, 1902). 



* ' Recherches sur quelques lois genSrales qui regissent les lignes et surfaces 

 algfibriques de tous les ordres,' Gerg. Ann., vol. svii. (1827), pp. 214-252. 



