Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 9)

530 
ON THE SCALENE TRANSFORMATION OF A PLANE CURVE. 
[617 
Reverting to the case where the locus of A is the circle 
this gives 
r'- — 2yr' cos 6 + 7 2 — /¿- = 0, 
t = 7 cos 6 4- V(/f 2 — T sin 2 6), 
1 _ 7 cos 6 — \/(/i 2 — 7 2 sin 2 6) 
r 7 2 — /t 2 
so that for the transformed curve we have 
r 2 + r 1 
in 
~)yeoae + r(l + 
! * 2 — 7 2 sin 2 6) + ni 2 — n 2 — 0. 
Putting for shortness 
ively, this is 
l 2 — m 2 
7 2 -IP 
= X, and for r, r cos 6, r sin 6, writing ^(a? + if), x, y respect- 
x 2 + y- + (1 — X) >yx + (1 + X) h 2 (x* + y 2 ) — 7 2 y 2 \ + m? — n 2 = 0, 
or, what is the same thing, 
[sc 2 + y 2 + (1 — X) yx + m? — n 2 ) 2 = (1 + X) 2 [IP {a? + y 1 ) — 7 2 y 2 \, 
a bicircular quartic. In the case X = —1, it reduces itself to the circle 
x 2 + y 2 + 2yx + m 1 — n 2 = 0 
twice, which is the case considered above; and in the case X = 1, or l 2 + h? = m 2 + 7-’, 
the equation is 
(x 2 + y 2 + m 2 — n 2 ) 2 = 4 [h? (x 2 + y 2 ) — 7 2 y 2 }, 
so that the curve is symmetrical in regard to each axis. In the case 7 = 0, the locus 
is a pair of concentric circles, centre B. 
The equation 
[x 2 + y 2 + (1 — X) yx -P m 2 — ?i 2 } 2 = (1 4- X) 2 {h 2 (x 2 + y 2 ) - 7 2 y 2 }, 
which contains the four constants X, 7, h and w? — n 2 , may be written in the form 
(a? + y 2 4- Ax + B) 2 = ax 2 4- ey 2 , 
(where the constants A, B, a, e are also arbitrary). This is, in fact, the equation of 
the general symmetrical bicircular quartic, referred to a properly-selected point on the 
axis as origin, viz. the origin is the centre of any one of the three involutions formed 
by the vertices (or points on the axis); say it is any one of the three involution- 
centres of the curve. 
To show this, assume 
(x — a)(x — /3) (x — 7) (x — 8) = x? —px 3 + qxr — rx + s:
	        
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