O'S THE THEORY OF POINT-GROUPS. 69 



on a curve of order m will all cut this curve again in the same nm - ((«)) + 2 

 points. This is a fallacy, for since, by hypothesis, the ({n))—2 arbitrary 

 points lie on a curve of order m, which, since ((«)) — 2 >((??i)) — l when 

 n>m, would not have been possible in the original theoi'em without further 

 conditions, it is now possible that the system of curves of order »i should 

 consist of degenerate curves, viz. the given curve of order in together 

 with a system of curves of order n—m which all pass through the addi- 

 tional arbitrary point, which point, therefore, taken in combination with 

 the {(n)) — 2 arbitrary points, fails to determine uniquely a curve of order 

 n, and the line of argument adopted in the original theorem falls to the 

 ground. (The correct statement in such a case is that the curves of order 

 n all cut the curve of order m again in an iiifinite number of points.) But 

 the exact number of arbitrary points which may be assumed upon the 

 curve of order m without invalidating the previous line of argument can 

 be found as follows : It is clear that, since n>in, the system will always 

 contain certain degenerate curves, each of which consists of the given 

 curve of order m, and some fixed curve of order n — m. Such a degenerate 

 curve can play the part assigned to U=0 in the original theorem, and 

 all the curves of order n through the ((w)) — 2 arbitrary points must pass 

 through ((w— 3)) additional points on it ; it only remains to decide how 

 the arbitrary points are distributed between the two curves of which it is 

 composed, and what distribution of the additional points will then result. 

 Now the conditions of the problem require the degenerate curve to be 

 fixed, and this can only be efiected by means of the assumption on the 

 curve of order n — m of a sufficient number of the arbitrary points to deter- 

 mine it completely, i.e. of ((« — «i)) — 1 ; the remaining arbitrary points 

 which are {(n)) — '2 — {{n—'m))-{-\ in number, lie upon the given curve of 

 order m ; and the difference between nm and the last-named number, 

 viz. ((?)i-3)), is the number of additional fixed points in which all the 

 curves of order n will cut the given curve of order m again. 



Two equivalent algebraical statements of Pliicker's original theorem 

 are given in a paper which he published in Crelle's ' Journal ' in 1836.^ 



(I.) Si Von donne a deux quantitcs variables successivement {(p)) ~2 

 coujjles de valeurs quelconques, et si Von suppose que ces valeurs satisfassent 

 a une equation quelconque d%i n'"'" degre entre les deux variables, il y axira 

 n- — ((n)) — 2 = ((n — 3)) couples de valeurs nouveaux qui safisfont a la nieme 

 equation et qui dependent uniquement des couples pre'cedents. 



(II.) Si Von connatt ((n)) — 2 couples de racines de deux equations du 

 n"""' degre entre deux inconnues, Von obtiendra les ((n — 3)) couptles de 

 racines restanfes, sans avoir recours a ces equations. 



In the same paper the new theorem is also stated in algebraical form : 



Si Von connait nq — ((q — 3)) couples des racines de deux e'qications du 

 n"""' et du q"'""' degre entre deux inconnues, n e'ta'id 2^lus grand que q et 

 qphis grand que 2, Von en deduira les ((q — 3)) coup)les des racines restantes 

 sans recourir aux equations proposees, en fonction des racines connues et 

 par la resolution de deux equations du ((q — 3))"""' degre. 



It is established as follows : 



Let n^p + q, and let ((;j)) — 1 of the {{n)) — '2 couples of values 

 which in the original theorem were all arbitrarily assumed, be now assumed 

 to satisfy an equation A^, = of order p, which is then completely 



' ' Th6oremes g^neraux concernant les equations d'un degre quelconque entre un 

 nombre quelconque d'inconnues,' CreJle, vol. xvi. pp. 47-57 ; Worlis, pp- 323-333. 



