178 



Prof. Cayley, On the transformation 



[Oct. 28, 



New Members of Council. 



Professor Humphry. 

 Professor Cayley. 

 Mr W. M. Hicks. 



The foHowing communications were made to the Society : — 



(1) Professor Cayley, On the transformation of co- 

 ordinates. 



The formulae for the transformation between two sets of 

 oblique co-ordinates assume a very elegant form when presented 

 in the notation of matrices. I call to mind that a matrix denotes 

 a system of quantities arranged in a square form 



( a , /S , 7 ), 

 '^■', 13', 7 



see my "Memoir on the Theory of Matrices," Phil. Trans, t. 

 cxLVii. (1857), pp. 273 — 312; moreover (a, f3,^\x, y, z) denotes 

 (,x-{- /3j/ + yz, and so 



( a , /3 , -f \x, y, z) 



a' , ^' , 7 



a", /3", 7" 

 denotes {ax + l3y + yz, -Ix + /3'y + y'z, ax + ^'y + y"z), 

 and again 

 ( a , /S , y \x, y, z\^, ij, ^ denotes ^ {a x + ^ y + y 



+ 'r}{rx x + ^' y + y' z) 



^' , /3' , y 



II all " 



^ , P , 7 



Consequently 

 ( a , /3 , 7 \x,y,z\^,'r],^) = {a, a , a" J^, rj, ^\x, y, z) 

 a', /S', 7' /3, /3', /3" 



a", iS", 7" 7. 7'. 7" 



In the case of a symmetrical matrix 



(a, A, <7 ) , 



h, h, f 



9' /. c 



