116 Proceedings of Royal Society of Edinburgh, [feb. 21, 
(or |- powers) of the central temperatures; and therefore inversely 
as the --‘ jK - , or , or powers of the diameters. 
k - I 2 — k 6 
(3) Under still the same conditions as (1) and (2), the quantities 
of matter in the two masses are inversely as the 
powers (inversely as the cube roots) of their diameters. 
(4) The diameters of different globular gaseous stars, of the same 
kind of gas, and of the same central densities, are as the square roots 
of their central temperatures. 
(5) The diameters of different globular gaseous stars of different 
kinds of gas, but of the same central densities and temperatures, are 
inversely as the square roots of the specific densities of the gases. 
(6) A single curve [^=/(^ -1 )] with scale of ordinate (r) and 
scale of abscissa (y) properly assigned according to (18), (17), and 
(11) shows for a globe of any kind of gas in molecular equilibrium, 
of given mass and given diameter, the absolute temperature at any 
distance from the centre. Another curve {[y with scales 
correspondingly assigned, shows the distribution of density from 
surface to centre. 
It is easy to find, with any desired degree of accuracy, the 
particular solution of (13), for which 
u = A, 
and ^ = A', where x = a 
ax 
. . (19), 
a denoting any chosen value of x, and A and A' any two arbitrary 
numerics, by successive applications of the formula 
u 
»+i 
A- f dxi A' - f*dx-) 
J a \ J a PJ 
( 20 ); 
the quadratures being performed with labour moderately propor- 
tional to tbe accuracy required, by tracing curves on “ section ’’-paper 
(paper ruled with small squares) and counting the squares and parts 
of squares in their areas. To begin, u 0 may be taken arbitrarily ; 
but it may conveniently be taken from a hasty graphic construction 
by drawing, step by step, successive arcs * of a curve with radii of 
* This method of graphically integrating a differential equation of the 
second order, which first occurred to me many years ago as suitable for find- 
ing the shapes of. particular cases of the capillary surface of revolution, was 
successfully carried out for me by Prof. John Perry, when a student in my 
