62 ON TRILINEAR CO-ORDINATES. 



cheres Mesphres Thothmosis, and in sucli a variety of ways that no 

 liypotliesis of accidental coincidence can account for the similarity; 

 third, that the incestuous birth of Adonis, together with the names of 

 his parents, find an exact parallel in the history of Thothmosis ; and 

 finally, that, as representing the Sun-god as beloved of Venus, descended 

 from Apollo, and of the line of Hermes, he reproduces the son of the 

 Sun, favourite of Athor, the son of Horus, and prince of the line of 

 Thoth, Acencheres Mesphres Thothmosis. 



ON TRILINEAR CO-ORDINATES. 



BY JAMBS LOUDON, M.A. 



MATHEMATICAL TUTOR AND DEAN, UNIVERSITY COLLEGE, TORONTO. 



The following method of treating the properties of the straight line 

 occurred to me in 1867, since which time I have used it with advantage 

 in the Lecture Room : 



1. To find the equation of the line ^ B' C which cuts the sides 

 BC, CA, AB of the triangle of reference in the points A', B', C 

 respectively. 



Let the angles at A', B', C be denoted by 6, ^, d', respectively; 

 then if P be a point in the line between A' and B' of which the 

 trilinear co-ordinates are a, /3, y, 



A'C'-PB^ -f- B^C( -PA' = A'Bf-PC 

 A^C ^ . B^C^ A^B^ 



em ' em sirn// ' 



Similar relations hold for different positions of P. 



If now the convention be made that a, /?, y are, respectively, negative 

 according as P lies between B' and C", C" and J.', A' and B', and 

 positive in all other cases, the above relations may be written 



B'C , C^A^ „ , A'Bf 

 sin 6 sm sin i// ' 



or 



Za + m/3 -f ny = 0, (1) 



•which is the required equation. 



The signs of (1) will evidently depend on the position of P. 

 2. From (1) it follows that 



I sin 6 m sin ^ n sin i^/ 



B'C' C'A' ~ A'B' 



(2) 



