350 SECTIONAL TRANSACTIONS.—A*. 
THe INTEGRATION OF EMDEN’S EQUATION AND THE APPLICATION OF 
NuMERICAL RESULTS :— 
Mr. D. H. Sapter.—The Numerical Integration of the Equation. 
A short account is given of the computation of numerical solutions of Emden’s 
Equation, and the precautions taken to ensure a solution correct to seven decimal 
places. The method used involves the calculation of differential coefficients at each 
point and the application of Taylor’s theorem. The advantages of the application 
of the method to this particular problem are described. The results are compared 
with Emden’s original solutions. 
Prof. EH. A. Mitne, F.R.S.—Functions associated with Solutions. 
In the application of Emden’s differential equations to problems of stellar structure 
certain equations occur (called ‘equations of fit’), which determine the nature of 
configurations describable in terms of solutions of Emden’s equations of different 
indices. These can be represented concisely in certain cases by equations 
u(a) = U(b) 
5 8 
1a) —Vi(b)ipe = 
(a)= Vib) x 5 
where $8, and (, are stellar parameters, U and V are subsidiary functions associated 
with solutions of Emden’s equation of index n = 3,and wv and v are subsidiary functions 
associated with solutions of Emden’s equation of index n = 3. For any index n, 
w and v (or U and V) are connected by a first order differential equation 
udv utv—l 
vdus ss u+ mv —3 
wand v being defined by 
ae &0” = t0’ 
See te 
where 1 d/,,d@0 : 
pa (ae) =~ 
and § = 6(&). 
The object of the investigation is to determine the roots a and b. The complete 
solution is given for all types of solutions of the two Emden equations concerned, 
partly from analytical considerations, partly from numerical quadratures by 
N. Fairclough. When U and V are derived from an Hmden solution of n = 3, there 
are either no solution, one solution, two coincident solutions or two distinct solu- 
tions according to the type of the solution of m = 3 used for u and v and the values of (8, 
and §.. These results are important in the theory of stellar structure. 
Mr. N. FarrcLoven.—Numerical Results for Indices 3 and §. 
For a given value of the index, the integrals calculated depend on one parameter. 
Tables are of two kinds, those in which individual integral curves are followed, and 
those which show the dependence on the parameter of some associated function 
or of some magnitude, such as the abscissa of the first zero or of the first maximum, 
which has not more than one value for each integral. 
Mr. R. H. Fowtsr, F.R.S.—Methods of Studying Emden’s Equation. 
A short account was given of certain methods which can be used with success in 
analysing the nature and arrangement of the solutions of equations of the type 
6” + 270" =0 
especially as  —> co*¥and as > 0. This equation is equivalent to Emden’s equa- 
tion when o = — 4. 
* See p. 566 for references. 
