EEV. T. P. KIKKMAN ON LINEAK CONSTBUCTIONS. 289 



whose relation to the surface we are examining, to the 

 form 



H{Y^+^Y,a:-»H-fY,a--»+ 4. ^^Yx + fYJ=rO, 



where H is a factor that may be disregarded, and Y Yj Y,... 

 are lengths from o on the axis of y, and ^ is an arbitrary 

 length on that of oj. If now n-1 of the points that define 

 the surface are on the axis of ^, the w* point in which that 

 axis meets the surface, is found by drawing a small number 

 of additional parallels. 



The expression before us — which, like the paradigm of 

 which it is the reduction, is a purely geometrical datum 

 and proposition — is an equation whose co-efficients and 

 roots are not numbers but lines. Since the addition and 

 multiplication of lines are subject to the same laws of aggre- 

 gation and commutation with those of numbers, there is no 

 reason in the world why the doctrine of equations, to a 

 certain point at least, should not be property common both 

 to arithmetic and geometry ; so far, namely, as the symbols 

 in the theory of equations retain their perfect generality. 

 By this theory, we know that S, the sum of the roots of 

 the above equation, is given by the proportion 



~S:|=:Y,:Y, 



and — S is found by the drawing of two parallels. It has 

 been already shown that the addition and subtraction of 

 lines from lines on the axis of x can be effected by the 

 ruler; hence the sought point is obtained by subtracting 

 from -j- S, thus constructed, the known sum of the other 

 n-1 roots of the equation. 



All that is proved above concerning loci of the iV* order 

 holds for constructions in loci of the N^ class ; if the co-ordi- 

 nates, X y z, &c., determine not points, but lines or planes. 

 If the axes of the line or plane co-ordinates are parallel 

 axes, as they may be if we so choose, I think it will be 

 found that all the constructions in question will be effected 



2p 



