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equation to the concrete equation (1). On comparing these two sets 

 of equations, we see that x and y in the first are involved in precisely 

 the same manner as a and b in the second, so that if any equation 

 F(w, v) = Q were given, it might determine one or the other by pre- 

 cisely the same algebra, according as x, y or a, b were substituted 

 for u, v. Whence flow Pliicker's theories of collineation and 

 reciprocity. 



The investigations respecting the concrete equation, on the other 

 hand, are altogether new. The most general form of the two 

 equations, is 



Concrete, OR=/;O, y) . OA+/ 2 (>, y) . OB 

 Abstract, 0(X y)=0, 



which will clearly determine a curve as definitely as before. If in 

 lieu of the abstract equation (#, y)=0, we were given the locus of 

 the two equations 



OM=a? . OA+3/ . OB, 0(#, y)=0, 



and from any point M in this curve we drew MN parallel to OB, 

 cutting OA produced in N, so that ON=#.OA, NM=y .OB, we 

 could find x and y from this curve, and consequently form 



OL=/;(#, y) . OA, and LR=/ 2 <>, y) . OB, 



and by this means determine the point R, where OR=OL + LR, in 

 the locus of the general equations, corresponding to the point M in 

 the particular locus. The general form is therefore the algebraical 

 expression of a curve formed from another curve by means of ope- 

 rations performed on the coordinates of the points in the latter, as 

 when an ellipse is formed from a circle by altering the ordinates in a 

 constant ratio. 



The algebraical treatment of this case consists in putting 



and between these equations and (#, y) = 0, eliminating x and y, 

 to find \l/ (p, <?)=0. The locus is then reduced to that of 



OR=^ . OI + q . OJ, $ (p, )=(), 



which is the ordinary simple case. But the whole of this latter 

 locus does not in all cases correspond to the locus of the general 

 equations, because not only x and y, but also^? or/, (a?, y), and q or 



