44 
a smith’s prize paper, 1877. 
[645 
The lines in the neighbourhood of a point of maximum, or of minimum, parameter 
are ovals surrounding the point in question, each oval being itself surrounded by the 
consecutive oval. Supposing that there are two points of minimum parameter, we 
have round each of them a series of ovals, until at length an oval belonging to 
the one of them comes to unite itself with an oval belonging to the other, the two 
ovals altering themselves into a figure of eight. Surrounding this we have a closed 
curve (in the first instance a deeply twice-indented oval) which (in the case supposed 
of there being, besides the two points of minimum parameter, a single point of 
maximum parameter) is in fact an oval surrounding the point of maximum parameter, 
and the remaining curves are the series of ovals surrounding that point. If we 
project stereographically from the point of maximum parameter (so that this point 
is represented by the points at infinity) we have a figure of eight, each loop 
containing within it a series of continually diminishing closed curves, and the figure 
of eight itself surrounded by a series of continually increasing closed curves. 
8. The investigation by means of the Potential presents the difficulty that the 
Potential of the infinite cylinder has no determinate value, as at once appears from 
the limiting case where the cylinder is reduced to a right line ; the difficulty is 
perhaps rather apparent than real, inasmuch as the partial differential equations 
dV d 2 V dV 
contain only differential coefficients , where as representing an attraction, 
and therefore also 
d*V 
dr 2 ’ 
are determinate. But it is safer to work directly with the 
Attraction; the Attraction of an infinite line acts in the perpendicular plane through 
the attracted point, and is inversely proportional to the distance; the problem is 
thus reduced to the plane problem of a circle of uniform density, force varying as 
(distance) -1 , attracting a point in its own plane. This is precisely similar to the 
case of a sphere with the ordinary law of attraction; dividing the circle into rings, 
each ring exerts an attraction = 0 upon an interior point, and an attraction as if 
collected at the centre upon an exterior point. Hence, writing a for the radius of 
the cylinder, and r for the distance of the attracted point, the attraction is = 7rr 
ira, 
for an interior point, and = for an exterior point. 
9. The theory is precisely the same as for curves ; taking the surfaces to be 
IT = 0 of the order m, and V=0 of the order n, the general form of the equation 
of a surface of the order r (r not less than m or n) is LU + MV = 0, where L is 
the general function of the order r — m, and M the general function of the order 
r—n; and so long as r is less than m + n, we obtain the required number of 
arbitrary constants as the sum of the numbers of the coefficients of L and of M, 
less unity. But as soon as r is =m + n a modification arises, viz. we obtain here 
an identity by assuming L = V, M = — U, and so for any larger value of r, we have 
an identity by assuming L = V(f>, M = — U(f), where </> is the general function of the 
order r—m — n. 
10. The numbers are known to be 1, 2, 4, 4, 2, 1, which values are obtained most 
easily (though not in the way which is theoretically most interesting) by finding for