14 Trans. Acad. Sci. of St. Louis. 



In the general ease we have a = 'I'' ( R), X = tp (/?), but as 

 the forms of these functions are very complicated it is not 

 easy to evaluate this integral except by some convenient pro- 

 cess of mechanical quadrature. In his paper on the Theoreti- 

 cal Temperature of the Sun, Lane has developed X in a con- 

 verging series which enable us to find its numerical value for 

 any argument with moderate facility. It is evident that at 

 the center of the sun X = <r , and at the surface \= 0. 



Now suppose we express the density of the shell in units 

 of the mean density of the sun, which is about 1.4 that of 

 water, and from a table of X; in which R, is the argument. 

 Then Lane's work shows that X } will vary from X Q = 20.06, 

 at the center, to X; = 0, at the theoretical upper limit 

 of the solar atmosphere. On the same basis *, will vary 

 from 20.06 to 1. The function a^ is therefore finite and 

 continuous from R — 0, to R = R t , at the surface of the 

 sun. If the sun's radius be divided into i equal parts, the 

 functions ^ may be computed by the formula: 



%\ + °i\ + °A + •••• + 0i\ 



_ / /v -r v^ t " 2 /*a -r t "t" t ^ 2 2 ) 



where , x iy . .0 i 2xe the volumes of the central nucleus, 

 and of the successive shells by which it is surrounded ; X* be- 

 ing their several densities and Vi the volumes of the corre- 

 sponding enclosing spheres. In the case of the sun it was 

 deemed sufficient to divide the radius into forty equal parts ; 

 the following table gives the values of these several functions 

 as determined by computation. 



