700 Lord Kelvin : The Problem of 
numbers, bnt he represented his solutions by curves *. He 
did give some of his numbers for three points of each curve, 
and Mr. Green, by very different methods of calculation, has 
found numbers for the case k = 2*5, agreeing with them to 
within T ] Qth per cent. 
§ 26. By improvements which Mr. Green has made on 
previous methods of calculation of Homer Lane's Function, 
and which he describes in an Appendix to the present paper, 
he has calculated values of the function & K (x), and of its 
differential coefficient ©'*(#), which are shown in five tables 
corresponding to the following five values of k, 1*5, 2*5, 3, 
4, co . For the four finite values of k the practical range of 
each table is from x = q to x = vo , q denoting the value of x 
which makes £ = 0. 
§ 27. There is such a value of x which is real in every case 
in which tc is positive and less than 5. This we see exem- 
plified in the four diminishing values of q found by Mr. Green 
(•2737, -1867, -1450, -0667) t for the four finite values of «, 
1*5, 2*5, 3, 4, and in the zero value of q for k = 5, the 
case described in § 29 below. In this case equation (24) has 
a solution in finite terms, which gives t "= VZ.x for infinitely 
small values of x, and therefore makes q = for x = 0. 
§28. Two interesting cases, a:=1 and k=5, for each of 
which the differential equation (24) is soluble in finite terms, 
have been noticed, the former by Ritter J, the latter by 
Schuster §. Bitter's case yields in reality Laplace's cele- 
brated law || of density for the earth's interior (smnr/r), 
which Laplace suggested as a consequence of supposing the 
earth to be a liquid globe, having pressure increasing from 
the surface inwards in proportion to the augmentation of the 
square of the density. With Ritter, however, the value of n 
is taken equal to 7r/R, so as to make the density zero at the 
bounding surface (r = R). With Laplace, n is taken equal 
to f-7r/R to fit terrestrial conditions, including a ratio of 
surface density to mean density which is approximately 
1/2*5. The ratio of surface density to mean density given 
by Laplace's law, with w = |7r/R, is in fact 1/2*4225, which is 
as near to 1/2*5 as our imperfect knowledge of the surface 
density of the earth requires. 
* American Journal of Science, July 1870, p. 69. 
t See Appendix to the present paper, Tables I. . . IV. 
% Wiedemann's Annalen, Bd. xi. 1880, p. 338. 
§ Brit. Assoc. Report, 1883, p. 428. 
|| Mecanique Celeste, vol. v. livre xi. p. 49. 
