Equations by Factors and Differentiation. 39 



minimum and the lower does not, and that when only one ca- 

 tenary can be drawn it does not correspond to a minimum." 

 According to these results it would seem that the problem does 

 not always admit of a continuous solution. To prove that this 

 conclusion cannot be true, it suffices to remark that it is possible 

 to draw through the given points any number of continuous 

 curves which by revolution about the axis would generate sur- 

 faces of different magnitudes — and that, as the decrement of 

 magnitude from one surface to the next inferior cannot go on 

 unlimitedly, a limiting minimum surface must eventually be 

 reached, the magnitude of which the Calculus of Variations 

 ought to be capable of determining. Consequently there must 

 be some fault or defect in the usual process of solution. It is 

 true, as Mr. Todhunter has pointed out, that a discontinuous 

 solution may be deduced from the equation y=.cs/\ -\-p* ; for if 

 c=0, and jo be not infinite, y = 0, and, c being zero, if p be 

 infinite y has arbitrary values. Hence the conditions of the 

 problem might be satisfied by the two ordinates to the given 

 points and the portion of the axis of x between them. But this 

 discontinuous solution affects in no manner the validity of the 

 foregoing argument for a continuous solution. 



The difficulty thus encountered will, I believe, be found to be 

 completely removed by the new process of integration I have 

 proposed, which rests on the principle that if a differential equa- 



d¥ d¥ 



tion containing F and — vanishes when F = and -—=0, an 



integral of that equation of an order lower than that of F is an 

 integral of F = 0. In the present instance the function F is 



i yq 



Vl+p 2 (1+p 2 )*' 



and it has been shown that the equation (ft), which belongs 

 to any involute of the evolute of the catenary obtained by inte- 



d¥ 

 grating F = 0, is satisfied if F = and j~ = 0. Consequently, 



by the rule just stated, such involute is an integral of F = 0, and 

 may be employed for solving the proposed problem . On account 

 of the additional arbitrary constant contained in this integral, it 

 is always possible to make the involute pass through the two 

 given points. 



The other problem is thus stated in Chapter V., art. 73, of 

 the work already cited : — " To determine a solid of revolution 

 the surface of which is given, so that it may cut the axis of revo- 

 lution at given points and have a maximum value." The solu- 

 tion I am about to propose would be applicable if the generating 



