ON THE THEORY OF POI^T-GROUPS. 89 



(0), (1), itc. ; they are written, e.g., (0"-il), (0"-"! 11 224). The ' facteurs- 

 seconds ' are connected with the ' facteurs-premiers ' by the following per- 

 fectly definite statement, in which is seen the importance of the numerical 

 notation : ' Chaque chiffre du facteur-premier annonce, dans le facteur- 

 second qui lui est joint, \\r\e puissance des lettres a, b, c, etc., dont ce chiffre 

 est I'exposant, et ces puissances sont autant de termes qu'il y a de 

 maniere de les arranger.'^ In other words, each ' facteur-second ' is a 

 symmetric function of the roots a, b, c, . . . of dimensions equal to the 

 sum of the numbers contained in its bracket. ' Si done les racines de 

 I'e'quation A etaient connues, il serait aise d'avoirtous les facteurs-seconds 

 de I'equation C. Mais ces racines sont inconnues, lorsque I'equation A 

 est d'un degr6 trop eleve pour que I'algebre en puisse donner la solution. 

 Cependant ces facteurs-seconds se peuvent toujours calculer et exhiber 

 sous une forme rationnelle, au moyen des coefficients de I'equation A.' '-^ 

 And then from the known connection between the coefficients of an 

 equation and the products of the roots taken one, two, . . . together. 

 Cramer, by a somewhat intricate rule, deduced the values of all the 

 symmetric functions required for the determinations of equation C. 



The essential feature of this rule is an arrangement of the 'facteurs- 

 premiers ' in lines according to a certain law : the first line,'* for example, 

 being of the type (0000), (0001), (0011), (0111), (1111), the second and 

 third lines are (0002), (0012), (0112), (1112); (0003), (001 3), (01 13), 

 (1113), and so on. The 'facteurs-seconds' corresponding to the first line 

 are easily seen to be these said sums of the products of the roots taken 

 one, two, three, four together, and are thus equal to 1, [1], [1-], [1'], [1*]. 

 All the ' facteurs-seconds ' corresponding to the succeeding lines are then 

 shown to be derivable, by means of a multiplication theorem for pairs of 

 ' facteurs-seconds,' from ' facteurs-seconds ' corresponding to terms in lines 

 above them, and thus, finally, from the first line ; they can therefore be 

 expressed rationally in terms of the coefficients of A. A typical example 

 of the multiplication theorem is : "' 



(0"-'111223)x(0"-il)=(0"-nil224)-f2(0"-nil233) 



+ 3(0"-'*112223)-f4(0"-nill223) 



where, as Cramer is careful to point out, the sum of the significant figures 

 in eacli bracket of the result is the same, being equal to the sum of those 

 in the two factors. The proof of the theorem lies in translating the 

 symbols into the actual terms for which they stand, viz., using the modern 

 2 notation, 



'S.abcdhJ^ X :i.a=-$abcdyf* + ^^ahcd^e^P + ^%abcH''e-^j^ + i^abcde'fhf 



as is sufficiently evident. A formal proof could, of course, easily be obtained 

 by induction. When the coefficients of equation A are, as we have already 

 assumed them to be, functions of y, the above characteristic of the multi- 

 plication theorem enables Cramer to prove that the degree in y of the 

 equation C cannot exceed mn. For each ' facteur-premier ' is easily seen 

 to contain y to a power {mn — sum of its numbers), and it only remains to 

 show that the corresponding ' facteur-second,' when evaluated in terms of 

 the coefficient of A, will contain y to a power equal to the sum of the 

 numbers in its bracket. 



Now this is seen to be true of the ' facteurs-seconds ' corresponding to 



• Loc. cit., p. 664. ' Loc. C'it,, p. 665. ' Zoc, rAt., p. 6G3, '' loc. ctt.. p. 668. 



