38 REV. HUGH MARTIN ON TRILINEAR CO-ORDINATES I 



fine vein of mathematical truth opens up in reference to them, which it needs 

 only a little ingenuity to work advantageously. Thus, at a glance, it is evident 

 that if a point moves in the straight line la+m (3+ny=0, its inverse moves in 



the locus, — + -a + — =0; which is a conic passing through the angular points of 

 ci p y 



the triangle of reference. Since writing the following pages I find Mr Townsend 



has a chapter entitled " Theory of inverse points with respect to a circle ;" and 



although not treated according to the trilinear method, these points, so called, 



will be found, I rather think, if so treated, to present only a case of what I have 



called "inverse points" in general. In the same manner I have called two lines 



" inverse" with respect to each other when the co-efficients of the co-ordinates 



are respectively the algebraical inverses of each other. 



There is another relation between a special point and line which I have not 

 ventured to designate, but to which I would respectfully call attention as requir- 

 ing designation. When lines from the angular points of a triangle are drawn 

 through any point to intersect the opposite sides, the intersections constitute 

 the angular points of an inscribed triangle, whose sides are known to meet the 

 corresponding sides of the original triangle in points which range in a straight 

 line. Instead of giving a particular designation to this line, I have used the 

 general functional symbol ; and, as its position depends exclusively on the point 

 — say P, I have called the line <p (P), in a few theorems in reference to it (Theorems 

 XXXI. — XXXV.) . Of course the inverse functional symbol (p ~ l indicates the point 

 in reference to the line, as the direct symbol indicates the line with reference to 

 the point. This point and line are, indeed, with respect to each other, a species 

 of pole and polar, — the line being the ordinary polar, not of the point but of its 

 inverse, — to the imaginary conic a 2 +/3 2 +7 2 =0. Manifestly a special designation 

 is necessary in a case like this, in order to secure that ease of reference and that 

 brevity of treatment without which the pioneering work of farther investigation 

 is brought to a stand. 



Theorem LXVI. is the prize question of the " Gentleman's Diary" for 1841; and 

 some long but good geometrical demonstrations of it are given. The proof is per- 

 fectly simple according to the trilinear method, and the co-ordinates of the point 

 appear in a form so elegant that one could not help seeing that it must have some 

 singular relations and be worthy of a name. I have accordingly ventured to call 

 it the Anapole of the two given points ; and, connecting it with some of the pre- 

 ceding results, I find a few propositions easily deducible, such as that the anapole 

 of two inverse points and the line joining them are pole and polar, to the imaginary 

 conic, a 2 + j3 2 + y 2 = 0. For the three concluding theorems I am indebted to a young 

 mathematical friend — destined, I believe and trust, to scientific eminence — Mr 

 George M. Smith, student in the Aberdeen University. On proposing to him the 

 problems of finding the locus of the anapole of a central body and its planet, and 



