290 EEV. T. p. KIRKMAN ON LINEAB CONSTRUCTIONS. 



by the drawing of given converging lines ; so that a finite 

 ruler will in general solve the problems. 



I readily allow that these linear constructions, although 

 they will, as I flatter myself, be found rigorously geometri- 

 cal, are far from being reduced to their most simple form ; 

 and I could state, if space were allowed me, methods of 

 abridging the operations indicated. If the locus under 

 consideration be represented not by Cartesian co-ordinates, 

 but by a system of terms, each being a product of linear 

 functions of w y and 2:, which represent distances from given 

 lines or planes measured in a determined direction, a slight 

 modification of th& above method of using the ruler, will 

 often bring out the required result with comparatively little 

 labour. I shall content myself with one example of a more 

 compendious method, forming the solution of a problem of 

 remarkable interest; to find the ninth intersection of two 

 curves of the third order through eight given points. 



The principal difficulty in the solution of the ninth point 

 problem, lies in the finding linearly a fifth known point on 

 each of the two conies through the points 12349 and 82349, 

 (vide page 83 of the 6th vol of the Cambridge and Dublin 

 Mathematical Journal), 



[123458] [123467] (67) (58) — [123457] [123468] (68) (57) 

 = 0=z (12349), 



[123458] [823467] (67) (51) — [823457] [123468] (61) (57) 



zzOzz (82349) ; 

 where [123458] is the integral function of the co-ordinates 

 of the six points 123458, which vanishes if they are on a 

 conic; the aconic function [12345 8'], as it has been denomi- 

 nated by Sir W. R. Hamilton, and (67) =1 o, denotes the 

 integral equation of the line through 6 and 7. 



A fifth point on the conic (12349) is the intersection of 

 the two lines 



[123458] (58) — [123457] (57) = o rz m 



[128468] (68) — [123467] (67) = o = v. 



