534 
ON THE SCALENE TRANSFORMATION OF A PLANE CURVE. 
[617 
s fc X 
we have s = kr, s + s' = r + r', that is, r = ^, r' — — s + s'. Substituting in the equation 
between r, r', written for greater convenience in the form 
(r + r') (rr 4- on 2 — l 2 ) + (l 2 — n 2 ) v — 0, 
the relation betAveen s, s' is found to be 
(s + s') s 2 + y + m 2 -14- {l 2 - n 2 ) s + s'^j= 0. 
On account of the term in s 3 , this equation in its general form does not, it would 
appear, give rise to transformations of much elegance. If, however, l = n, then the relation 
becomes 
(k — 1) s 2 + kss' + k 2 (m 2 — P) = 0 ; 
and in particular, if k = 2, then 
s 2 + 2ss' =4 (l 2 — m 2 ), or say (s + s') 2 — s' 2 = 4 (P — on 2 ), 
viz. taking A instead of b as the fixed point, the relation between the radii AC, Ab 
is p 2 — p' 2 = 4 (P — m 2 ); the cell is in this case Sylvester’s “quadratic-binomial extractor.”