PROFESSOR CAYLEY ON POLYZOMAL CURVES. 71 



where w, v, w are any trilinear co-ordinates whatever ; and take the inverse 

 co-efficients to be (A, B, C, F, G, H) (A = be — / 2 , &c), then for any given 

 point the co-ordinates of which are (u , v , w ), the equation of the tangents from 

 this point to the conic is, as is well known, 



(A, B, C, F, G, H)(y a w — iv v, iv to — u Q w, u v — v Q u) 2 = ; 



consequently for the conic 



(a, b, e,f, g, h)(u, v, wf = , 



where (u, v, w) are areal co-ordinates referring, as above, to any three given points 

 A., B, C, the equation of the pair of tangents from the point- 1 to the conic is 



(A, B, C, F, G, IT) (i -*z,i- Sz, % - 7 zf = , 



and that of the pair of tangents from J is 



(A,B, C, F, G, H )(„- a'z, q - pz, r, - y'z) 2 = , 



these two line-pairs intersecting, of course, in the foci of the conic. 



169. In particular, if the conic is a conic passing through the points A, B, C, 

 then taking its equation to be 



Iviv + rmvtc + nuv = , 



the inverse co-efficients are as (P, m 2 , n 2 , — 2mn, — 2nl, — 2lm), and we have for 

 the equations of the two line-pairs 



Jl{% - «z) + Jm(% - fc) + Jn{i - yz) = , 

 */l(ri — a'z) + >Jm{ri — Wz) + J n{n — yz) = ° ■ 



Tlie Theorem of the Variable Zomal — Art. No. 170. 

 170. Consider the four circles 



A° = 0, B° = 0, C° = 0, D° = (A° = {x- az) 2 + (y - a'z) 2 - a'h 2 , &c), 

 which have a common orthotomic circle ; so that as before 



aA° + bB° + cC° + dD° = , 

 where 



a : b : c : d = BCD : - CD A : DAB : - ABC . 



I consider the first three circles as given, and the fourth circle as a variable 

 circle cutting at right angles the orthotomic circle of the three given circles ; this 

 being so, attending only to the ratios a : b : c, we may write 



a : b : c = DBC : DC A : DAB , 



that is, (a, b, c) are proportional to the areal co-ordinates of the centre of the vari- 

 able circle in regard to the triangle ABC. 



