116 The Right Hon. Lord Kelvin [May 12, 



a determinate finite solution if the offered land is an oasis surrounded 

 by valueless desert. It has also a determinate finite solution even 

 though the land be nowhere valueless, if the wall is sufficiently 

 more and more expensive at greater and greater distances from some 

 place where there are quarries, or habitations for the builders. 



The simplified case of this problem, in which all equal areas of 

 the land are equally valuable, is identical with the old well-known 

 Cambridge dynamical plane problem of finding the motion of a par- 

 ticle relatively to a line of reference revolving uniformly in a plane : 

 to which belongs that considerable part of the " Lunar Theory " in 

 which any possible motion of the moon is calculated on the sup- 

 position that the centre of gravity of the earth and moon moves 

 uniformly in a circle round the sun, and that the motions of the 

 earth and moon are exactly in this plane. The rule for curvature 

 which I have given you expresses in words the essence of the cal- 

 culation, and suggests a graphic method for finding solutions by 

 which not uninteresting approximations * to the cusped and looped 

 orbits of G. F. Hill f and Poincare J can be obtained without dis- 

 proportionately great labour. 



In the dynamical problem, the angular velocity of the revolving 

 line of reference is numerically equal to half the value of the land 

 per square yard ; and the relative velocity of the moving particle is 

 numerically equal to the cost of the wall per lineal yard in the land 

 question. 



But now as to the proper theorem of curvature for each case ; 

 both Dido and Horatius Codes no doubt felt it instinctively and 

 were guided by it, though they could not put it into words, still less 

 prove it by the " calculus of variations." It was useless knowledge 

 to the bees, and, therefore, they did not know it ; because they had 

 only to do with straight lines. But as you are not bees I advise you 

 all, even though you have no interest in acquiring as much property 

 as you can enclose by a wall of given length, to try Dido's problem 

 for yourselves, simplifying it, however, by doing away with the 

 rugged coast line for part of your boundary, and completing the 

 enclosure by the wall itself. Take forty inches of thin soft black 

 thread with its ends knotted together and let it represent the wall ; 

 lay it down on a large sheet of white paper and try to enclose the 

 greatest area with it you can. You will feel that you must stretch 

 it in a circle to do this, and then, perhaps, you will like to read 

 Pappus (Liber V. Theorema II. Propositio II.) to find mathematical 

 demonstration that you have judged rightly for the case of all equal 

 areas of the enclosed land equally valuable. Next try a case in 



* Kelvin, "On Graphic Solution of Dynamical Problems." 'Phil. Mag.' 

 1892 (2nd half-year). 



| Hill, " Researches in the Lunar Theory," Part 3. National Academy of 

 Sciences, 1887. 



1 ' Me'thodes Nouvelles de la Mecanique Celeste,' p. 109 (1892). 



