[From the Prootediityf of the Can&ridge Philosophical Society, Vol. IL, 1876.] 



LI 1 1. On Problem in the Calculus of Variations in which the solution 



is discontinuous. 



THE rider on the third question in the Senate-House paper of Wednesday, 

 January 15, 1J to 4, was set as an example of discontinuity introduced into 

 a problem in a way somewhat different, I think, from any of those discussed 

 in Mr Todhunter's essay*. In some of Mr Todhunter's cases the discontinuity 

 was involved or its possibility implied in the statement of the problem, as when 

 a curve is precluded from transgressing the boundary of a given region, or wheiv 

 its curvature must not be negative. In the case of figures of revolution con- 

 sidered as generated by a plane curve revolving about a line in its plane, this 

 forms a boundary of the region within which the curve must lie, and therefore 

 often forms part of the curve required for the solution. 



In the problem now before us there is no discontinuity in the stateu)t-nt. 

 and it is introduced into the problem by the continuous change of the 

 efficients of a certain equation as we pass along the curve. At a certain point 

 the two roots of this equation which satisfy the minimum condition coalesce 

 with, each other and with a maximum root. Beyond this point the root which 

 formerly indicated a maximum indicates a minimum, and the other two roots 

 become impossible. 



* Rtttarthft in the Calculus of Variations, ttc. 



[The question referred to was set in 1873, and is as follows : If the velocity of a carriage along 

 roftd i proportional to the cube of the cosine of the inclination of the road to the horizon, determine 

 the jmth of quickest ascent from the bottom to the top of a hemispherical hill, and shew that it consist* 

 of the spherical curve described by a point of a great circle which rolls on a small circle described about 



the pole with a radios ~ , together with an arc of a great circle. How is the discontinuity introdnnil 

 into tliu problem 1] 



