286 



REV. T. P. KIEKMAN ON LINEAR CONSTRUCTIONS. 



What we have done in the plane of y 2^ by the aid of our 

 »-uuit t,, we can imitate by drawing pairs of parallels, in 

 the plane o£ x y, by the aid either of jj or ^; for Xo ei, x^ «„ 

 &c., are given rhomboids in that plane, having each an 

 angle at O. Three pairs of parallels will effect the trans- 

 formation, 



Azz H^xl . e . Ci . x^t . Xrje^ . x^i zz t^i^ . icj . ecj . XjXjXj, in which 

 ofsCg zz 9}x„ Xffii zz JJX2, x^i zz 9)X3 ; and a fourth pair gives 

 A zz ^jj* . a^fl . Xi Xs X3 ; which, by three pairs more, and 

 the equations ex* zz ffj, ^5X3 ir ^e^, ^5X3 — ^y, becomes 

 A = l'rtle,^,i^^o zz ^n'^e,x^l zz l^ri'^Ya^ i 

 X1X2X3 being lengths cut off from O on the axis of a?, and e^ e^ 

 and Y lengths on that of y, where Y is positive or nega- 

 tive with A. Thus, by drawing twelve pairs of parallels, 

 we effect the reduction 



Azzioi^,!/* ,!^ .xy ,yz.zx ,x .y,z .Izz Cv*^ . x*o, 



t a S3 44&S6;89 



The term, B :r: — a;2 . y? . 2? • ara^s . y^^ . z^x^ - Xe-y^'Zj.l^ 

 zz ^>3*^'Y^ a^. is reduced in the same manner, Y, being a 

 length from O on the axis of y, and positive or negative 

 with B, the sign of which depends on the co-ordinates by 

 which that solid is determined. 



Let B be supposed positive,- then the addition 



A + B=:^^^f (Y + Y,)«i 

 is to be performed. If c and c^ be the extremities of Y and 

 Y, remote from O, and be the line intercepted between the 

 axis that cuts off Y r;z Og ; draw cb, parallel to the axis of 

 a? to b, in b b, parallel to that of y; join bc^j and parallel to 

 this draw b^ e^ to c^, in the axis of y; the length Oc„ n Y -}- 

 Yi. This requires the three lines bc^ cb, be,,, be and bb, hav- 

 ing been drawn before. The subtraction ofY, from Y might 

 have been performed with equal facility, giving Y-Y^, a 

 length cut off from O. 



The sum of all the terms of the paradigm which contain 

 xi» can thus be reduced to a single term ^|^»;* K wl, in which 



