SECTION III., 1907. [89] Trans. R. S. C. 
IX.—A Certain Type of Isoperimetric Problem, in particular, the Solid 
of Maximum Attraction 
By Mr. Norman R. Witson, Winnipeg. 
Presented by Professor Alfred Baker and read May 15, 1907. 
INTRODUCTION, 
In the fourth chapter of his ‘‘ Methodus inveniendi lineas curvas 
maximi minimive proprietate gaudentes,” Euler has given an ingenious 
method of transforming isoperimetric problems of a certain class to the 
non-isoperimetric type. In the first chapter of this paper we discuss in a 
general way the conditions implied in the transformations for Euler’s 
examples, and the circumstances in which it is effective in removing the 
isoperimetric condition in other isoperimetric problems.! In the second 
chapter we consider the transformation in detail for the solid of revolu- 
tion of maximum attraction, after giving a brief critique of the partial 
solutions that have already been given of this problem. We obtain the 
form that the solid must take in order to furnish the maximum attrac- 
tion, and show that this actually does produce a maximum. The latter 
result depends upon certain relations between the senses of description 
at the points of intersection of a straight line with a simple closed curve 
of a special class. In the third chapter we establish these results of 
Analysis Situs. 
CHAPTER I. 
EuLER’Ss METHOD OF REMOVING THE ISOPERIMETRIC CONDITION. 
$1. The Curve of Least Arc and Given Area. 
(a). The first of the problems which Euler has solved by the 
method above referred? to, he enunciates: ‘‘ Supra axe, AP, construere 
lineam, BM,ita comparatam ut abscissa area, AB MP, datae magnitudinis, 
arcus curvae, BM, illi respondens, sit omnium minimus.” Kuler does not 
explicitly state whether the end-point, M, is free to move in any way or 
not. Axes of x and y being chosen as in Fig. 1, the area, u, swept out 
1See also Kneser Math. Annalen, 59, and Erdmann, Crelle’s Journal. 
2 See pp. 143-4, 
