397 



obvious extension of the notation, and by altering the order of the 

 factors, 



[0 1 , 2 M-:-0 2 , 1 M].[0 2) 3M^03, 2 M]...[0 W)1 M-f-0 1 , n M] 



=(-i)*. [0,0',-rO.oy . [O 2 o' 3 -o 3 o' 2 ] . .. [O.os^doy 



= (-!). [(_!). (_l). ... (n factors)... (-I)"] 

 = (-!). (-!)-=(- 1).(-1)=(-1)*.= 1*. 



9. Hence if all the roots are scalar, that is, if each side of the 

 polygon cuts the scalar radical locus in as many points as its abstract 

 equation has dimensions, we have the following fundamental propo- 

 sition of the theory of transversals : 



" If we take the points at the extremities of a side of a polygon, 

 and find the ratios of the distances of these points from the intersec- 

 tions of that side with a curve, and proceed in this manner regularly 

 round the polygon in one direction, the product of all these ratios 

 will be unity." 



10. Thus if the triangle O^O, cut a straight line in A, B, C so 

 that these points lie on O 2 O 3 , OjOj, (^O, respectively, then 



2 O.A 0,5 



O 8 C ' O 3 A ' O.B 



If the same triangle OjOjOg cut a conic in the points A lt A 2 by O 2 O 3 i 

 B lf B 2 by O t O 3 ; C lf C 2 by O^ respectively, then 



2A A 2A 2 A 2 3 B 1 0,B 

 0,0, ' 2 2 X O.A, ' 3 A 2 X OA ' 1 B 2 



* Pliicker says, = +1, and adds, that " it is immediately evident that when the 

 curve is of an even order, the upper sign must be taken ; and when it is of an 

 uneven order, either the upper or the lower, according as the number of the angles 

 of the polygon is even or odd " (System, p. 44). Accordingly most of his exam- 

 ples, as shown in (10), give an erroneous result. It must be observed that 

 Pliicker does not give the equation in the form above written, but places the pro- 

 duct of all the antecedents of the ratios on one side, and the product of all the 

 consequents on the other. As, however, these antecedents and consequents are 

 directed lines, such products have no proper meaning. He has in fact considered 

 those directed lines as scalars, but has neglected to assign the directed unit lines 

 with respect to which they are to be considered as scalars. Hence his error. 



f Pliicker makes the second side of this equation = 1, which is manifestly 

 erroneous. But if OjO, O 2 , 3 c be drawn from Oj, 2 , O a through any point M 

 to O 2 3 , OjOg, OjOjj respectively, then by considering the intersections of the tri- 

 angles OjOjfl, OjOga with 3 e and O a i respectively, we immediately deduce from 

 the equation in the text, 



Ojc 2 3 g_ _j 

 <V ' O 3 a " Oj* 



