ftEV. T. P. KIEKMAN ON LINEAR CONSTRUCTIONS. 



295 



At F on the other an absciss having the sign of y, — y^. Draw 

 the ordinate FF^ meeting a?5 in F'', and next F^'Xj parallel 

 to the second diagonal to Xs in the axis of x. 

 then is k . O X = k. (0F'» + F"X) =: ^ . ^^x +'^x, 



i .6 13 4 6 6 1 



and (y • x—x—i/ x) (^yl^x^y x) (^y . x) — k' ox . OX OX 



Draw now from c a parallel to the third diagonal, meet- 

 ing the axis of x in G, and from the intersection G, of the 

 ordinate GGi, with oe^a draw Gjx' to x' in the axis of ar, 

 parallel to the diagonal of Icx^,,, the last of the five. We 

 have thus y^ . ce^,—x^, — Vf^x^,, =.h.{0 Gi+ Gix')=: ^. Ox*. 

 Next let a parallel to the first diagonal meet either Xy or x^ 

 on the axis of y, intercepting at H, on the other, an absciss 

 having the sign of y? — yg > ^^^ from H', the intersection of 

 the ordinate HH^ with a^j draw a parallel to the fourth dia- 

 gonal, to x/ in the axis of x. Then is Jc . Oxx' zr k . (OH 

 -f-H'xi') n: y^ — y^ . x^ — x^ + y* «?; and by drawing a parallel 

 to 24, and an ordinate meeting the axis of x in x'a, we 

 obtain 



(y,.a:e,-^a— ya/aJeXy?— ^2— ^4— ^5)_(y7— y.-«0=^- Ox', Ox',. Ox', 



By drawing, in addition to X ¥ before drawn, the four 



2 Xs, ¥X,, X,k\y k\X,, XeA-l, parallel to 24 like Xk\ k'Xe to 



2 Xj, andAjjXy to 2X2, drawn before, we effect the reduction, 



/P. OX . OX, OX, = F wi» OX; 

 Drawing next 2x} and 2xJ, and a broken line of four 

 strokes, beginning at x' and ending at Xj, we obtain ^. Ox\ 

 Ox}. Oxa zr A^ m\ Oxj; and the line to be drawn is now 



OX, . 51 — Oxs 57 = = w, . 

 Join 17; let 1 Xj meet 7 X7 inp'; let Op' meet/r, in q^; let 

 q^r^y parallel to 1 X5, meet 17 in r' ; 5 r* is the line required. 

 The point r' is found by drawing 32 lines additional to the 

 67 already drawn before. 

 The line v^ is 



(y.x^x—v.x)(y^ Jx^x + ]/x) (y^y . a;)(61) — (y. z^ — ya;) (y^. xH + yx) (y^*) (67)=o 



eiita ae 1346 61 81 4 Ciieiai aie|724a 67 874 



