334 
THE OUTSIDE OF A STAR 
since the boundary value of / is found to be . This replaces the first 
approximation acT 4 = J — H (2 + 3r). 
It will be noticed that we do not use the “revised” value of H from 
(230-4); that would be a retrograde step. That H is constant and equal to 
a given boundary value is the essential condition of the problem. The 
fact that, calculating from the first approximation to the temperature 
distribution, we do not exactly reproduce a constant H indicates the 
failure of the first approximation to fulfil the conditions of the problem; 
and if the “revised value” is H + AH, AH is a measure of the error of the 
first approximation. 
The functions TJ r (r) are calculated by the reduction formula 
(r - 1) U r (r) = e~ T — rU r _ x (r), 
and U 1 (r) is a tabulated function. 
In Table 45 the second column gives the value of / calculated from 
(231-2), the third column gives the ratio of the second approximation to 
T 4 from (231-3) to the first approximation from (226-5). The fourth column 
gives AH/H as defined above. 
Table 45. 
Second Approximation to Temperature Distribution. 
T 
/ 
AH/H 
0 
•8235 
•875 
•0000 
i 
•9188 
•967 
+ -0176 
i 
•9497 
•993 
+ -0241 
i 
•9755 
1-010 
+ -0263 
1 
•9915 
1-016 
+ -0194 
oo 
1-0000 
1-000 
•0000 
The main interest of the investigation is that the second approximation 
makes so slight an amendment to the distribution given by the first 
approximation. We could without much difficulty examine the effect of 
this amendment on the law of darkening (227-2) but it is evident that the 
correction would be trivial*. 
The first approximation will be sufficiently accurate for most purposes, 
and we shall use it throughout the rest of this Chapter. 
The Photosphere f. 
232. The photosphere is the region in which the heat and light directly 
reaching us is emitted. We shall take two levels r x and t 2 between which 
* The chief effect is a slight extra dimming near the limb, where the radiation 
comes from small depth r and would in the limit be reduced to of the intensity- 
given by the first approximation. 
f The investigation of conditions in the sun’s photosphere is resinned in § 251 
and reference should be made to that section for the definitive conclusions.