264 ON a smith’s prize question, relating to potentials. [757 
being four cubic curves. The whole curve (1) is shown in fig. 2, and the others are 
Fig. 2. 
equal or opposite curves: the rationalised equation of (1) is in fact 
2a 2 (y + a) 
X ~(y + ay-a>’ 
and by writing — a for a, and in each equation — x for x, we have the rational 
equations of the other three curves. 
But, drawing only the required portions of the curves, we have fig. 3 exhibiting 
Pig. 3. 
the potentials of the two hemispheres A and B; and also the discontinuous potential 
of the whole shell, the ordinate for this last being the sum of the ordinates for the 
two hemispheres respectively.