fiEV. T. P. KIRBLMAN ON LINEAE CONSTRUCTIONS. 293 



parallel to 2X, and 2X2; k, and k„ being in the axis of y, and 

 X' and X" in that of a. Then since m: OX=z OX : OX', 

 and m : OX' = OX, : OX ", OX . OX, . OX, = w» OX", 



and (3/0 . a?e — a?a — ^y» a^o) (^2 • a?6 — ^^g) ?/8 — '^i . a;* =: P m* OX"; a 

 transformation effected by 15 applications of the ruler, when 

 the Cartesian co-ordinates, and the four lines determining 

 the points a e Cy have once been drawn. 



Let now 57 and 71 be drawn to c, and e, : let the diagonals 

 of k cc,^ — x^^ h x^ — x^, like-signed with x^^ — x^ and x^—^Xj, and 

 that ofk x^, of the sign contrary to that of a;,^ be drawn. Four 

 parallels and two ordinates, after the manner of c A, AiX, 

 2X„ OiB, AAi BXi above, will suffice to determine the 

 points X Xi X2 such that 



(y . X — X — yx) Cy . X — x) (y — y . x) 11: A'. Ox. Ox Ox ; 



0/6/* a 0/867714 18 



and by drawing five lines after the manner of 2X2, X^j, ^-jX^, 

 X'^^,, It^Xl' above, we shall obtain the point x" on the axis of 

 x such that 



(y . X — X — ya:) (y . x — x^ {ju — y . a;) rz J^m^OTi!'. 



0% ae3S7714 



II I 



consequently the line to be drawn is 



OX." (08) ~ Ox" 57 = o rz «. 



Join 78; let 8x" meet 7X" inp; let Op meet Xj in qi let 

 qr parallel to 8x" meet 78 in r ; then 5r is the line required. 

 This is constructed at the expense of 40 applications of the 

 ruler, besides the drawing of the Cartesian co-ordinates of 

 the eight points. 



It is necessary in the next place to find the line 



iV .x^ — yx)(p . x^) (f:^ .X) (6S) — (9'X^—yx){y.x^ (jPya:) (67; = o =» 

 deb be368 814 deb be867 714 



111 11 



where 6 = 12.46; d= 23.68, e =z 34.81 

 d — 23.67, e, r= 34.71; 

 By drawing, in addition to the lines, 46, 68, 67, the dia- 

 gonals of k jc, — ajb, k Xe — Xa, with proper signs, then tliree par 



