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XXXIX. — On Centres, Faisceaux, and Envelopes of Homology. By Rev. Hugh 

 Martin, M.A., Member of the Mathematical Society of London, and Examiner 

 in Mathematics in the University of Edinburgh. Communicated by Professor 

 Kelland. 



(Read 1st April 1867.) 



One of the theorems of a paper which Professor Kelland did me the honour 

 to read to the Society, in March 1865, opens up a field of geometrical investigation 

 so interesting and fertile, that I venture to ask attention to some of the results of 

 a partial examination of it in the following series of propositions. I think it right 

 to explain, that I do not venture to expect attention to them on account of any 

 importance attaching to them individually, but on account of their number and 

 somewhat elegant relations. Considered individually, they may be of little 

 importance, having no claim to rank, so to speak, among propositions of a 

 planetary magnitude. But a system of moons, however diminutive, may become 

 interesting if they present elegant relations among their mean motions and 

 longitudes ; and an orbit that would be grudged to a pigmy planet may be will- 

 ingly accorded to a host of planetoids. If this is still too exalted language in 

 which to speak of the following results, I can at least confidently affirm that 

 they indicate a direction in which an analyst of very moderate attainments may 

 easily discover for himself a shower of meteors. 



It is well known that when straight lines are drawn from the angles of a 

 triangle through any point in its plane, they intersect the sides in three points, 

 which form the angular points of a triangle so related to the first that the inter- 

 sections of their corresponding sides range in a straight line. As the triangles 

 are said to be in homology * we may conveniently designate the point as the 

 Centre of Homology, and the resulting straight line as the Line of Homology, — 

 the line represented in the former paper by 0(P X ). If we have a second point, 

 P 2 , inverse to the former, we have a second line of homology, which may be called 

 the inverse line. 



Now, if the centre of homology is subjected to motion according to a given law, 

 or in a given curve, the line of homology displaces itself, so as, in its varying 

 positions, to constitute a faisceau. The Envelope of this faisceau may then be 

 inquired for. Simultaneously the inverse centre of homology will move in a 



"" I fear this terminology may give an aspect of pretentiousness to the paper which is far from 

 being intended. But if I was to avoid the indefinite title, " On a Certain Class" &c, I confess I 

 could find no other sufficiently descriptive. 



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