294 REV. T. p. KIKKMAN ON LINEAR CONSTRUCTIONS. 



rallels and an ordinate, after the manner of c Ay AiX, 2Xi, 

 AA|, yre obtain two points X3X4 such that 



it/ . x^^ — yx) {y . x—x) (y^ . x) =1 k" OX OX OX. 



deb bea68 814 41 3 



Four lines more, setting out from X^, after the manner of 

 the four Xk,f hX', Xk^^ k^^X", will determine the point S on 

 the axis of x such that OXi OXz OX^ zz w' OS, or 



(y . X — X — yoc) {y . x — x) {y — y . x zzz khn'OS. 



deb be26 88 14 



Next, the co-efficient of (67) in v :zi o can be reduced by 

 drawing the diagonals of A (xl — x\), k (xe — a,), and then as 

 above three parallels and an ordinate, to A' Os^ Ox^ Ox^, which 

 four lines now will transform to A' m^ Os, s being a point 

 found on the axis of x. Thus twenty-three new lines are 

 expended in transforming the problem to the shape 



OS . (68) ~ Os (67) =:o=zv, 

 and r in the sought line 6r, is obtained by drawing other 

 four. Draw now 5r and 6r^ to meet in P; and we have 

 found a fifth point of the conic ( 12349) by 67 applications 

 of the ruler, if the Cartesian co-ordinates are drawn before- 

 hand. 



We have yet to construct the pair of lines 



C234581] (51) _ [234587] (57) = o = u, , 

 [234681] (61) — [234687] (67) =: o =: Vj . 



Now[234587]=zl>e(^e,-^aj+ye„(^a-^o;+yaX^o-a?e.)] 



.DS ai being (72*45), e„ being (34-87), and D^ being the 

 quantity 



(y-v . x-x — y~y . x-x) . (y-y x-x — y^ x^) .{y~y . x^ -— y^ x^'), 



83 58 68 2. 3 348 7 8734 45 72 7246 



Omitting from D and D^ the first of these three factors, 

 the line Mj zz is 



p. x-x— yx) (y-y . x-x + yx) (f^ . x) (51) — {y . x^-r yx) (y^.x^ + yx) (j^-y . x) (57)==o 

 c e a a e 1 2 4 5 5 1 8 I 4 o ei, ai a,e„ 7245 67874 



Draw 72 and 87, to find a and e on 45 and 34. Draw the 

 diagonals of ^ (Xi — iCs), kx^^ k Xc^^ — x^^, Iix„ having the signs of 

 these areas, and that of ^a^e,/ having its sign opposite to that of 

 Xeif. A parallel to the first of these five diagonals meeting the 

 axis of y on 07ie of a7i, 0:2, as the case requires, will intercept 



