Proceedings of Sections. 



Section A. 

 ASTRONOMY, MATHEMATICS, AND PHYSICS. 



1.— ON THE GEOMETRY OF AN AXIS OF HOMOLOGY. 



Part IH. 

 By EVELYN G. HOGG, M.A., Christ's College, Christchurch, New Zealand. 



This paper is a continuation of ttoo previous ones read before the A.A.A.S. 



at its Hobart (1902) and Dunedin (1904) meetings. 



§ 1. On related triads of points — 



Let the trilinear co-ordiuates of a point O referred to a triangle of 

 reference ABC be {0^/3^^^). The equation of the axis of homology of O 

 with respect to this triangle will be 



a 3 7 



° *o '^O ^o 



Tf A + ^ + 1/ = 0, the following six points will lie on Lq, viz. : — 



f^i(ao^. ^o/"' 7o")' f-M«o/'' /^o*'' 7o^)' ^3(«o»'' /^o'"^' 7oA0 

 Oi("o^' /^o'^ Vo/O' ^iia^u, ^^fx, 7o\), O^i^a^ti, fd^X, -{^i'). 



The three Q points in which X, /i, u are permuted in cyclical order 

 and the three O points in which A, v, u are permuted in cyclical order 

 each form what is called a related triad. Such triads will be referred to 

 as the triads (\ ^t »'), {\ i> /<). 



When the co-ordinates of a point are (oq'^- /^o/'' 7o'')' ^^^^ point will 

 be called the point (\, /u, v), if no confusion appear likely to arise from 

 the omission of the multipliers a^, (3^, 7q. 



The symbols (X. O), (T. P) mean respectively the axis of honiology 

 of the point Q and the tangent at the point P. 



Let there be taken the two associated conies — 



s = -° + '^ + -° = 0. 



The axes of each of the triads {X /n v), (A, v ;t) form a triangle in- 

 scribed in S and touching 2. 



Let (X.o,)(X.C)3); (X,03)(X.Q,); (X.Q,) (X.fi^) intersect in the 

 points A, B, C, respectively. Then the co-ordinates of these points will 



be respectively (^{^ ^^ ;) (^' ^ -,) (j' 1'^). B^, C,A,. A,Bi will 



touch S in the points (\-, /t^ li') (^tr, v'\ X^) {v"-, X^ /<^) respectively. Thus 



from the triad {X ^i v) lying on L^ we derive a triad ( ^ 7I 7 ) b'i^g on the 



conic S and a triad (X^ /i^ i^^) lying on the conic 2. 



