350 Transactions. 



From the above relations we easily obtain 

 QiSi = ± 2APia, 



Q3S, = + 2AP3yi, 

 where Sj z^ «Ayi + ^yiai + cai/3i, 



therefore «, : A : y, = g-^ : g^ : gi-, (ii) 



The signs of the ratios following those of the co-ordinates 



Also "^^ = + 2Aaa, 



-^^^^=+2a^A 

 P2 



^%^^=+2ACy,. 



hence S, (^ + ^' + ^) = 2 a (aa, + &/3, + cy, ) 



' = 4a^ 

 If (ai/3iyi) move on a circle of radius (^ concentric with the circum- 

 circle, Si = ^ (E^- d^), and therefore 



E 



aQi , ^Q2 , cQa _ 4 A E 



P, Po Po E2 - ^2 



Hence the theorem " If P be any point within the triangle ABC at a 

 distance d from the circum-centre, and if Q be its isogonal conjugate, then 



BC ■ AQ CA . BQ AB . CQ _ 4aE 



AP + BP ^ ""CP E2-^2 ^"V 



If P lie outside the triangle ABC, the relation (iii) must be modified 

 in accordance with relation (ii), and (^^ — E^ substituted for E^ — d^. 



Art. XLVI. — The Action of Phosphorus on Solutions of Copper Sulphate 

 and certain other Metallic Salts. 



By H. Eands, M.A. 



Communicated by Dr. W. P. Evans. 



[Bead before the Philosophical Institute of Canterbury, 6th November, 1912.] 



Introduction. 

 A PIECE of ordinary yellow phosphorus when placed in a solution of copper 

 sulphate becomes covered with a black film of copper phosphide and then 

 with a firm coating of bright metallic copper. The black phosphide is 

 always found between the phosphorus and the copper. This reducing 

 action of phosphorus was made practical use of before 1865 in the 

 manufacture of phosphor-copper, but the course of the reaction was first 

 definitely investigated by Walter Straub in 1903. He concludes* that 



* Joura. Chem; Soc, 1903. vol. 84. ii, 593. 



