Prof. Cayley on the Theory of the Evolute. 345 



ordinates, the equation of the tangent is 



LX+MY + NZ = 0. 



The coordinates of the pole of the tangent are therefore 



(A, H, GlL, M, N) : (H, B, FJL, M, N) : (G, ¥, C^L, M, N). 



And the equation of the normal is 



X , Y , Z 



* , y , * 



(A, K, GjL, M, N), (H, B, FjL, M, N), (G, ¥, ClL, M, N) 



The formula in this form will be convenient in the sequel ; but 

 there is no real loss of generality in taking the equation of the 

 absolute to be x* + y 2 + z' 2 = ; the values of (A, B, C, F, G, H) 

 are then (1, 1, 1, 0, 0, 0), and the formula becomes 



X, Y, Z =0; 



x, y, z 



L, M, N 



where it will be remembered that (L, M, N) denote the derived 

 functions 0*U, <3#U, "d z V), 



The evolute is therefore the envelope of the line represented by 

 the foregoing equation, say the equation 11 = 0, considering 

 therein {x, y } z) as variable parameters connected by the equa- 

 tion U=0. 



As an example, let it be required to find the evolute of a conic; 

 since the axes are arbitrary, we may without loss of generality 

 assume that the equation of the conic is xz — y' i = 0. The values 

 of (L, M, N) here are (z, — 2y, x). Moreover the equation is 

 satisfied by writing therein x:y:z=l:d:d 2 ; the values of 

 (L, M, N) then become (# 2 , — 20, 1) and the equation is 



= 0; 



i o , e* 



x , y , z 



(A, H, &X4, ~1) 2 > (H, B, F£0, -l) 2 , (G, F, CX#, -1 J 



or developing, this is 



X/ G0 3 -2F6> 2 + C0\ 



^_H0 4 + 2B<9 3 - F0 2 ) 



+Y( A0 4 -2H0 3 + G<9 2 \ 



V - G0 2 + 2F<9-C7 



+ Z/ H0 2 -2B0 + F\ 



V _A0 3 -|-2H<9 2 - GO /=0, 



which I leave in this form in order to show the origin of the 

 Phil. Mag. S. 4. Vol. 29. No. 197. May 1865. 2 A 



