WILSON | CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 49 
(b). Referring again to Kuler’s first two examples, (§§ 1, 2), in 
order that the area between the arcs and lines shall have an arithmetic 
meaning, we must suppose the curve made up of these has no double 
points. In the first case, for example, thisis ensured by the fact that BM 
is of type, y = f(x), and y > 0, conditions assumed for other reasons. 
On the other hand, if we use parameter representation, permitting the 
curve to turn back on itself, the inequalities equivalent to the non-exist- 
ence of double points may be expected to take a somewhat complicated 
form in the transformed problem. The first question to be settled, there- 
fore, is whether or no there exists in the transformed ret of curves a 
variation of the type required in the fundamental lemma. Again, if this 
question can be answered in the affirmative, the transformed double point 
condition may be expected to play an important role in proving that the 
extremal so discovered actually furnishes a minimum. 
Similar remarks apply to any problem in parameter representation 
in which there is a distinction between inside and outside points. Our 
purpose is to discuss these questions in detail for the solid of revolution of 
maximum attraction. 
CHAPTER EL. 
THE SOLID OF MAXIMUM ATTRACTION. 
S1. Historical. 
As stated in the last paragraph, our object in the present chapter is 
to apply this method of Euler’s in detail to determine the form of the 
solid of maximum attraction. More explicitly, given a quantity of 
homogeneous matter, bounded by a surface of revolution and attracting, ar- 
cording to Newton's law, to find the form of the generating curve in order 
that the attraction upon a particle on the axis of revolution and in contact 
with the surface shall be a maximum. 
(b). The problem is first mentioned by Gauss,! (1830), in a paper 
on capillary attraction, where it is stated without proof that the maximum 
attraction is to that of a sphere of equal mass as 3 : 4/25. 
(c) The first discussion of the problem seems to have been given 
by Airy.” He takes the attracted particle as origin for a system of 
rectangular co-ordinates, the «—axis coinciding with the axis of revolu- 

1 Gauss, Ges. Werke, V,s. 31. 
* Airy, Math. Tracts: p. 309; Airy’s solution is reproduced by Jellett (1850, 
Variations, p. 307), Todhunter (1871: Researches, p. 120), and Carll (1881 : 
Variations, p. 141). Todhunter computes also the second variation for I + 2 K, 
which is negative. 
Sec. III., 1907. 4 
