14 On a supposed Failure of the Culculus of Variations. 



The first of these denotes a sphere of radius —2a, the first or 

 last limit upon the axis of x being- arbitrary. The second de- 

 notes a cylinder whose radius is indefinitely small. And the 

 union of the two, which gives the complete solution of the pro- 

 blem, is a sphere of such a radius that its surface has the pre- 

 scribed value, connected by indefinitely small cylinders or pipes 

 with the points adopted as the limits of x, that is, with A and B. 



The following diagrams may be conceived to represent the 

 various forms which the solution takes. The first is peculiar to 

 the case when the diameter of the sphere which has the given 

 superficies is less than AB; the second is peculiar to the case 

 when the diameter is greater than A B ; the third and fourth both 

 apply to both cases. The diameter of the small pipe is made 

 finite, to be visible to the eye. 



Form 1. Form 4. 



Form 2. 



Form 3. 



The practical solution evidently is, that a sphere is to be con- 

 structed, at any part of the axis of x, whose diameter is such that 

 its surface will be equal to the surface prescribed in the data of 

 the problem. But the metaphysical solution, containing the 

 idea of the tubular connexions with A and B, enables us also to 

 satisfy the condition of terminating the integrations at any points 

 that we may select, not necessarily defined by the position of the 

 sphere. 



Royal Observatory, Greenwich, 

 June 5, 1861. 



