3.2 Prof. Challis on a new Integration of Differential 



would seem, therefore, to follow that the condition of a minimum 

 can always be satisfied by a catenary so situated. But on trial 

 this is not found to be the case. Mr. Todhunter states (' Hist, 

 of Calc. of Var/ art. 308) that this problem was the subject of 

 an essay by Goldschmidt which obtained a prize in 1831, in 

 which "the conclusion is, that sometimes two such catenaries 

 can be drawn, sometimes only one, and sometimes no catenary." 

 The same problem is discussed at considerable length in arts. 

 102-105 of the Leqons de Calcul des Variations by MM, Lindelof 

 and PAbbe Moigno, and a similar conclusion is arrived at. In 

 fact, this appears to be a discontinuous solution of the problem. 



Yet it is certain that there can always be drawn between the 

 given points continuous curves which, by the revolution about 

 the axis, generate surfaces of different magnitudes; and as the 

 descending gradations of magnitude cannot go on indefinitely, 

 there must be a limiting minimum surface, the form of which 

 should admit of being determined by the Calculus of Variations. 

 The purpose of the following investigation is to show how this 

 may be done. 



First, it may be remarked that the present problem differs 

 from Problem I. in the respect that kdx is not, as well as Kpdx, 

 an exact differential. This is one reason for concluding that 

 A^ = and A = may not be used indifferently as if they were 

 equivalent equations. Another reason is, that the above first in- 

 tegral of Ap = does not satisfy A = and Ap = in the same 

 manner. For by substituting cVl+p* for y in the latter 

 equation, we have 



_ _^ . PQC _ 



Vi+p 2 1+p* ' 



which is satisfied if p = 0, but not if p = infinity; and by sub- 

 stituting the same value of y in A = 0, we get 



- 1 - qc =0, 



\/l+p* \+f 



which is satisfied if p — infinity, but not if p = 0, the value of c 

 being arbitrary. These considerations point to the conclusion 

 that the solution of the problem can be effected only by taking 

 into account an independent integration of A = 0. 



Such an integration I proceed now to obtain by a method 

 which, as far as I know, is new. The method consists essen- 

 tially in first finding, when it is possible, the evolute of the curve 

 or curves of which A = is the differential equation, and then 

 employing the involutes thence derivable, which may be regarded 

 as the solution of the equation, to satisfy, either by computation 

 or by graphical construction, the given conditions of the problem. 



