a Gas under its own Gravitation only. 291 



(2) Under the same conditions as (1) (that is, H and T the 

 same for the different masses), the central densities are as the 

 /cth powers (or -| powers) of the central temperatures ; and 



therefore inversely as the ^ , or ~ — 7, or ~~ , powers of the 



diameters. K ~ L Z ~~ k 6 



(3) Under still the same conditions as (1) and (2), the 

 quantities of matter in the two masses are inversely as the 



(x — 7 — 3)th 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 gaseoas 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 [,y=/(^ -1 )] w ^ n sca l e 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 mole- 

 cular equilibrium, of given mass and given diameter, the abso- 

 lute temperature at any distance from the centre. Another 

 curve, {y= [/0'~ 1 )] K } ? with sca^s 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 



du 

 u=A, and — =A / , where x=a . . (19), 



a denoting any chosen value of %, and A and A! any two 

 arbitrary numerics, by successive applications of the formula 



-= A -j>( A '-J>5) 



(20); 



the quadratures being performed with labour moderately pro- 

 portional to the 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 

 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 curvature calculated 

 from (13) with the value of dujdx found from the step-by- 

 step process. If this preliminary construction is done with 



* This method of graphically integrating a differential equation of the 

 second order, which first occurred to ine many years ago as suitable for 

 finding 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 



X2 



