192 Mr J. Brill, On Conjugate Functions [Feb. 27, 
The are of a Cartesian Oval makes equal angles with the right 
line drawn from the point to any focus and the circular are drawn 
from it through the other two foci. 
Taking the above figure, let S,, S,, 8, be the three foci of the 
curve, P a point on the curve, and P7' and PG the tangent and 
normal at P. Also, let PK be the tangent at P to the circle 
passing round the triangle S,PS,._ Then by Crofton’s theorem we 
have the angle 7PK =S,P7", and therefore S,PG=KPG. And 
we have KPG=S,PK + GPS,=PS,K+PS,K—PGK. Also 
S.PG = PGK — PS,K, and thus we obtain the angular relation 
2, PGK = PS,K + PS,K + PS,K. 
Hence we see that a system of confocal Cartesians form an equi- 
potential system; and, taking S, for origin, and writing S,S, =a 
and S.S,=0b, we see that the requisite function of a complex 
variable is obtained by solving the equation 
log “Y= — } log 2 — } log (2— a) — flog (2-0) + log f. 
Thus we have w= | ; le ) 
0 Jz (z—a)(z—b) 
which, if we write f=./b/2, is equivalent to z=asn’w, k= JalJ/b. 
3. The fact that logh—7S is expressible as a function of a 
complex variable is well known, but it does not seem to have 
been noticed that it is only the first member of a whole series of 
expressions of the same type. It is obvious that the operation 
that was performed upon w to obtain log h—7S may be repeated, 
and thus we shall be furnished with a new expression of a general 
type capable of being expressed as a function of a complex variable. 
If p, and p, be the respective radii of curvature of the curves 
£=const. and 7 =const. at their point of intersection, then 
! le and ao = ale 
Px 0€&’ Pe On 
Thus we have 
Caen (lacy 1 (A 
