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757] ON A smith’s prize question, relating to potentials. 
If (as may very well happen in particular cases) V is algebraically determinable, 
then, qua one-valued function of (x, z), V is not any root y at pleasure of an 
algebraical equation </> (x, y, z) = 0, but it is for any given values of (x, z), some one 
determinate root y 1 of this equation : and we thus see how in this case the before- 
mentioned discontinuity in the value of must arise: viz. when the trace of the 
plane meets the dark line the section is a curve having a double point; and, for 
the positions of P on the two sides of the dark line, we have V the ordinate 
belonging to different branches of the curve of section. If the path of P passes 
through an extremity of the dark line, then the curve of section will, instead of a 
double point, have in general a cusp; and when the path of P does not cross the 
dark line, then the curve of section is a continuous line without singularity. It may 
be added that the surface <£ (x, y, z) = 0 must have a nodal line which as to a certain 
finite portion thereof is crunodal, giving the before-mentioned double points of the 
sections, but as to the residue thereof is acnodal or isolated. 
It may happen that (the surface being algebraical) any particular section thereof, 
instead of being a single curve having a double point as above, breaks up into two 
distinct curves, so that for the two positions of P, we have V the ordinate of two 
distinct curves: and this is what really happens in the case of P a point on the 
axis of a circular disk or a spherical segment: thus in the case of the disk, taking 
c for the radius, and x for the distance from the centre of the disk, the formula 
is V= 2tt {V(c 2 + P-) ± x]; or writing V -r 2tt = y, the section is made up of the two 
distinct hyperbolas y (y — 2x) = c 2 , and y (y 4- 2x) = c 2 . 
It may be remarked that in each case, it is only for P on the axis that the 
potential is algebraical. 
In the case of the hemispheres, drawing OM a radius at right angles to the 
axis, the formula for the potential of an axial point P is of the form 
(PM - PA), 
or writing V = 2iry we have for the hemisphere A., the curve (1) or (2) accoiding 
as (x-a) is positive or negative; and for the hemisphere B the curve (3) or (4) 
according as x + a is positive or negative; viz. the equations are 
(1) y = - {V(« 2 + x2 ) — {%- a)}, 
CO 
(2) y — \ {V( a2 + ^ 2 ) + ( x — a )}> 
(3) y = - {\/(y + ^ 2 ) - (« + a)ji 
CO 
(4) y = - {V( a2 + ^ 2 ) + ( x + a)}, 
CO