ON THE THEORY OP rOINT-GROUPS. 71 



connection with the algebraical theorem called by its author's name, 

 a theorem which was afterwards destined to play an important part (at 

 the hands of Clebsch) in the interpretation of Abel's theorem into the 

 language of analytical geometry. 



Jacobi's method is, in fact, the very i^everse of Pliicker's. Tlie open- 

 ing paragraph of this memoir,^ after a brief reference to Euler's paper, 

 ' Sur une contradiction apparente dans la doctrine des lignes courbes,' ^ 

 and to the problem in the intersections of curves which is there dealt 

 with, states that those problems are of algebraical importance, and that it 

 appears advisable to investigate the equations of condition which exist 

 among the values of two variables which cause two integral functions to 

 vanish simultaneously. Throughout the course of this investigation the 

 arguments are strictly algebraical, although a geometrical equivalent of 

 each theorem is given. The following brief analysis of Jacobi's memoir 

 will show wherein his geometrical theorems differ in enunciation from 

 Pliicker's, although dealing with the same problems. 



The two integral functions which vanish by hypothesis for simul- 

 taneous values of the variable are, in the first place, to be of the same 

 order n, and in order to arrive at the number of equations of condition 

 ■which must exist among the values of the variables in this case, Jacobi 

 begins by considering a function u of order n which vanishes for ((«)) — 2 

 given systems of values. Since a function of order n contains {{n)) 

 coefficients (homogeneous), and since the given systems of values of the 

 variables provide ((«)) — 2 linear equations among the coefficients, it 

 follows that u can be written in the form a^a^^^x'y^ + b'Si>^^x'^)j'^, where a, h 

 are the two coefficients which are not eliminated from the system of 

 ((w)) — 2 equations linear in the coefficients, and a^g, b^g are the functions 

 of the {{n)) — 2 given values of the variables which, in solving for the 

 other coefficients, are the multiples of a and b respectively, a-f^ taking 

 all possible values from to n. Any other function v of ordei- n can 

 similarly be written as a'^a^.^x"}/^ + b'l,b^f,x''if , where a , b^^ are the same 

 functions as before. The common roots of ?i = 0, v=0 are seen to be 

 those of 2«.^.r'"«/^=0, 26„^.'«'"y''=0 and are n- in number, ((«)) — 2 of them 

 are already known, therefore the remaining {{^n — 3)) give rise to the 

 2((w — 3)) 'equations of condition' among the ni^ values of x and the w^ 

 corresiDondlng values of y, wliich are obtained by substituting them suc- 

 cessively in the two equations 2a^^:i;"_y®=0, 26, x"}j^^O. Hence the 

 theorem : 



Of tlie n^ systems of simultaneous values of x and y which satisfy two 

 equations of the xith order in x andj, ((n)) — 2 may be arbitrarily assumed 

 and tlie remaining ((n— 3)) are determined by these; or, among the n'^ 

 values of 'X. and the n- corresj^onding values of y there are 2((n — 3)) 

 equations of condition. 



The geometrical equivalent of this is : 



Of the n- 2'>oints of intersection of tioo curves of order n, ((n — 3)) are 

 determined by the rest. 



In the next section of his memoir Jacobi discusses tlie more compli- 

 cated case in which the two integral functions are of differing orders, 

 m, n. It is here that he makes use of certain {{m-\-n — 3)) equations upon 



' 'De relationibus, quaj locum habere clcbent inter puncta intersectionis duarum 

 curvarum . . . algebraicarum dati ordinis, simulcum enodatione paradoxi algebraici,' 

 Crelle, vol. xv. pp. 285-308 ; Works (Berlin, 1884), vol. iii. pp. 327-351. 



- Acad. Berlin, 1748, pp. 219-233; cf. § 8 of this Report (^Brlt. Assoc. Report, 1902). 



