KEV. T. P. KIRKMAN ON LINEAR CONSTEUCTIONS. 285 



dimensions ; yet he might not have been less qualified to 

 judge of the logic of tri-dimensional geometry, nor less pro- 

 fited by the devices which it employs for the solution of 

 problems in a plane. I beg the reader to believe, that to 

 the mathematicians in the planet Mercury, the outward 

 apperception of these entities is a very different affair Jfrom 

 what he finds it now. 



All that is incumbent on me is to show, that the propo- 

 sition 2 + al.yl.zl ,x^s . y^^ . z^x^ . ^g . y^ . xrg .I9 rz ex- 

 presses the law of the solid locus of the second degree. I 

 proceed to prove geometrically that every point x^y^^Zo^ which 

 by its co-ordinates satisfies the proposition, lies in a continu- 

 ous locus, such that no right line can meet it in one point 

 only, or in more than two. 



It is conceded readily, that in a product of lines, as in a 



product of numbers, the order of the factors is indifferent, 



so that 



x^ .y* . z^ .xy .yz .zx.x.y.zmx^.x.x.x.yz.yz.yz.yz. yzzziA. 

 1 a 33 44 55 6 7 8 3 6 6 I a 1 a 4 4 3 s 7 3 



Let O be the origin, and taking any three points on the 

 positive axes, ^v, y, z^ let 



Ox -I, Oy — n. Oz — l. 



The Cartesian co-ordinates being drawn, we have in the 

 plane of yz the parallelogram y^z^, having an angle at O. 

 Join the extremity of y, to z, the extremity of ^ ; and from 

 that of z^ draw a parallel to this joining line^ cutting the 

 axis of 2/ at a distance e from O. We have plainly Z, : z^ 

 zzyjie, or the parallelograms Z^e and y^^s are equal, e being 

 a length cut off from the origin on the axis of y, and positive 

 or negative according as y^ and z^ have like or unlike signs. 



By drawing four more pairs of parallel lines, after the 

 manner of the pair just drawn, we can reduce A to the form 

 A zz a^ . yf . 4 . x^yz . y^i . z^x^ .x^.yj . Zs=z xl. X3.Xi.X6 , 

 te . ^«i, . Z,e . Z,e3 . Zfit, e «i €2 €3 64, being lengths cut off from 

 O on the axis of y, and positive or negative as the case may 

 be, by the drawing of the described five pairs of parallels. 



