1907-8.] The Problem of a Spherical Gaseous Nebula. 293 
for each of which the direct verification of the solutions obtained is 
possible; the solution of the latter equation obtained was © 5 (x), which is 
given in § 6 above. 
In these cases, and in the case of any Homer Lane Function, if t 
throughout a number of intervals becomes greater than its true value, the 
d 2 t • dt 
absolute value of also becomes greater than its true value, and is 
(A/Jb (X/t K j 
therefore tending to become less than it would otherwise be. Hence the 
successive additions to t in each interval in which it is greater than its true 
value are, or tend to become, less than they would be with a correct t. 
Thus, when t has become too great in any interval, throughout the succeed- 
ing intervals it tends to return to its true value rather than to go without 
limit away from it. From similar considerations it may be judged that 
when t has become less than its true value, in the succeeding intervals it 
tends to return towards its true value. 
§ 10. The numerical values of the Homer Lane Function ©(&) with its 
differential coefficient in the interval from x=co to x = q, given in 
Tables I. . . . IV., and the values of the Boylean Function "Tpr), and the 
function ^ / (x)!' < ^(x), in the interval from x= ‘25, to x = T, given in Table V., 
have all been obtained by the method of § 8. In each of the Homer Lane 
Functions, a beginning of the calculation was made from the following 
approximate equation : — 
«■<">' 1 + H® 
( 12 ), 
easily derived from equations (2) and (4) above. From x = oc tox = '25, 
the Boylean Function was calculated by a method described below. 
After Tables I. . . . IV. had been completed, as it was still desirable to 
be able to verify the results obtained by the step-by-step process at some 
point close to the final point, q, and as the labour of calculating successive 
terms of the series (2) soon becomes very great, while the number of the 
terms required to give a sufficiently good result also becomes greater and 
greater as x diminishes, the form of expansion given in § 11 below was 
tried, and found to be useful. 
§ 11. Assume as a solution of equation (1), 
® K (x) = 
l + a i 
+ 
etc. 
( 13 ). 
We can write ®" K {x) in either of the following forms : — 
