1888. ] and Hquipotential Curves. 19M 
Next suppose that we have a lemniscate, having S and S’ for 
foci, and C for node. Let P be a point on the curve, and join 
CP, SP and S’P. Also, let PY and PG be the tangent and 
normal at P. Then, referring to the figure below, we have the 
angle SPG = CPS’, and therefore 
PSA + PS’A — PCA = SPC + PCA — CPS’ 
=CPG+ PCA 
= PGA. 
It is evident that a relation exactly similar to this will be 
satisfied by the angles which PT and the tangents at P to the 
circles passing through that point and having S’, C, S for centres, 
make with the line S’SA. Thus from the three systems of 
concentric circles having S’, C, S for centres we can deduce a 
system of equipotential curves having the lemniscate for one of 
their number. ‘To do this we have to solve the equation 
log oY = — log (z—a) —log (z+ a) +logz + logk, 
where & and @ are real quantities. This gives us 
cyt k 2 2 
=5 log (2? — a’). 
As a final example of this character, we will discuss the Cartesian 
Oval. Prof. Crofton gives the following theorem: 
oh / 
PSK 
“icf ° 
sh S, GAS; K T 
