282 REV. T. p. KIEKMAN ON LINEAR CONSTRUCTIONS. 



have changed the sign ; the latter is made from the 

 former by the transposition of the pair 4, and it will be 

 found deducible from the first term only by an odd number 

 of single transpositions. 



The paradigm, being formed, is either a purely geome- 

 trical datum, or an analytical one, according as we define 

 ^1 yi ^u <S;c., to be finite lines given in space, which we have 

 neither the intention nor the means of measuring, or num- 

 bers, as length, in feet, of the co-ordinates. If we consider 

 them for a moment to be numbers, the expression 



^-^^3^ ."i^ . !? . xy .yz , zx , X >y . z . 1 rr 



1 a 3314 55 678 9 



is the equation to the surface of the second order which 

 passes through the nine points 1, 2, 3,...9 ; for it is plainly 

 of the second degree in x^y^z^, and it vanishes when for x^y^z^ 

 we put the co-ordinates of any of the nine points. For let 

 {x^ y^ Zq) and (a*, y, zs) be the same point. The first term is 

 now identical in value, but not in sign, with the term 

 — ^ • y? • -sf • ^q^o • yt^i . ^6^5 .Xa.yj.Za I9, which being formed 

 fi*om the first by a transposition of the pair 3, stands in 

 the paradigm with a negative sign. And thus the whole 

 equation is reduced to a system of internecine pairs of terms. 

 We are thus in possession of a certain and surprisingly 

 simple method of writing down at once the equation to a 

 locus of any order (or class), in terms of the defining points 

 (or tangents), provided that their number does not exceed that 

 of the constants in the general equation of the iV^ order (or 

 class): and, what is of much importance, we have the equation 

 not only without the labour of elimination, but free from 

 all superfluous terms and factors ; while the most compen- 

 dious expressions which modern geometry employs to re- 

 present curves and surfaces, would, if written out at length 

 in terms of the co-ordinates included in the factors, be 

 generally found to contain an enormously disproportionate 

 amount of superfluous terms. 



