68 REPORT — 1903. 



XJ=0, V=0 so determined, then U + \V=0, where /\ is an undetermined 

 coefficient, is the equation of all those curves of order n which pass 

 through the n- points of intersection of the above two curves. It requires 

 one linear equation to determine X, and thus the knowledge of any new 

 point P on the locus U + W=0, but not on U=0 nor on V=0, is suffi- 

 cient for this purpose, and the equation of the completely determined curve 

 XJ-|-\,V=0 which passes through an arbitrary point P can be obtained. 

 Moreover, this same curve can also be uniquely determined by adding 

 a point Pto the ((n)) — 2 arbitrary points (since ((«)) — 1 points completely 

 determine a curve of order n), and it passes through the n'^ points of 

 intersection of U=0, V=0, i.e. through certain 9i-— ((?i)) + 2=((?i— 3)) 

 points common to U=0, V=0, as well as through the arbitrary points 

 and P. Now take another curve V'=0 instead of V=0, and obtain in 

 the same manner the equation U4-a'iV'=0 of a curve completely deter- 

 mined by the {{n)) — 1 points, and the same point P as before ; this curve 

 is therefore identical with U-l-/\iV=0 ; and it passes through certain 

 ((« — 3)) points common to U=0, V'=0. It has thus been shown that 

 XJz=0, V=0, U-f X 1^=0 all pass through certain {{n — 3)) points as well as 

 through the arbitrary points, and also that U=0, V'=0, U-H/uiV' = all 

 pass through certain ((«— 3)) points as well as through the arbitrary points ; 

 moreover, these points are in each case common to U=0 and to the 

 particular curve determined by the addition of P to the arbitrary points, 

 whose equation may be written either as U-|-XiV=0 or as U-f /uiV'=0 ; 

 that is to say, they are the same {{n — Z)) fixed points. By this argument 

 it can be shown that any curve which passes through ((n)) — 2 arbitrary 

 points cuts any other curve through these points in the same ((^i— 3)) 

 fixed points. 



The complete validity of this proof depends upon two assumptions : 

 that every curve of order n through the points of intersection of two 

 wiven curves U=0, V=0 of the same order has an equation of the form 

 XJ-f XV=0 ; and that a curve of order n is completely determined by 

 (^(^n))—\ points. The first of these is a very special case of a much more 

 «eneral theorem ^ which, so long as the method of counting the constants 

 of an equation was considered to aftbrd a sufficiently rigoi'ous proof of in- 

 formation obtained by its means, was supposed to be intuitively true. The 

 difficulties of a rigorous proof of the general theorem, moreover, do not 

 appear unless cases are considered in which the points of intersection are 

 multiple points on U=0, V=0, and the minute investigation of higher 

 singularities of curves had not yet been attempted ; it is not surprising, 

 therefore, that throughout Pliicker's lifetime the theorem in question was 

 taken for granted. With regard to the second assumption, the case is 

 difierent ; the paradox itself had arisen from a want of seeing exactly 

 how the element of indetermination could enter into the equation of a 

 curve drav.'n through ((«)) — 1 points ; and Pliicker, in the above proof, 

 expressly guards himself against exceptional cases by the use of the words 

 ' arbitrary,' ' in general,' etc. 



This iiad not, however, prevented Pliicker from previously (in 1828) 

 fallin<^ into a mistake which he afterwards corrected (in 1836). At the 

 end of the footnote to the problem of osculation he had stated, namely, 

 that the infinitely many curves of order n, n>m, through ((?i — 2)) points 



' Usually known as 'Noether's theorem.' See Math. Arm., vo\. ii. pp. 293-316 

 (1869) and Math. Atm., vol. vi. pp. 351-359 (1872). 



I 



