« 
[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 51 
(f). Kneser does not go beyond the consideration of the first 
variation, and a discussion of the sufficient conditions has never been 
given by means of the Calculus of Variations.’ Moreover, the previous 
treatments of the problem are open to certain minor objections. The 
assumption of the admissible curves in one of the forms: 
ANNONCE). 
involves a restriction which is not justified by the nature of the problem. 
Again, the conditions at the end-points, one of which is variable, have not 
been discussed. Further, the extremal, (3), furnished by the method of 
Airy, ceases to satisfy at the two points at which it meets 0 the con- 
ditions of continuity under which the general theory can be applied to 
the solution. The same remark applies to the solution (5), since at the 
Grn 
ae 
tion, but here another difficulty arises. A close analysis shows that the 
passage from the (r, #) to the (u, v) plane involves a number of restric- 
tions upon the slope of the curves in the (u, v) plane. As we have 
remarked in Chap. I, $6 (a), the question arises whether it is possible to 
secure à variation of the type required from this restricted set, a question 
which must be discussed in some detail; which we proceed to do. 
attracted particle coo- Kneser’s solution is not open to this objec- 
S2. Detailed Formulation of the Problem. 
(a). We use parameter representation to obtain the desired degree 
of generality. The attracted particle being chosen as origin for a system 
of rectangular co-ordinates, of which O x is the axis of revolution, we sup- 
pose that the meridian curve, £, is given in the form 
RP DT) UD) LOL TS Es USOT, 
where m and + areof class D’, meeting Ow at 7 = 7,, and 7 = 7,; and 
at these points only. We also assume that (ya’—a y") does not change 
: ee : - : A 
sign an infinitude of times; or in polar co-ordinates, that a does not 
iB 
change sign an infinitude of times. In other words, £, can be divided 
into a finite number of arcs on which, 
i ) @ is an increasing function of 7, or 
ii) @ is a decreasing function of 7, or 
ili) 6 is constant. 


1A sufficiency proof has, however, been given by Schellbach, (1851, Journal für 
Math., XLI, s. 343), by a method not belonging to the calculus of variations. By 
means of the force of attraction resolved along the axis of revolution a one-para- 
meter set of surfaces is obtained, each dividing space into a region of greater and 
less resolved attraction. For an absolute maximum the bounding surface must be 
one of these, the parameter being determined by the mass condition. 
