290 
Proceedings of the Royal Society of Edinburgh. [Sess. 
preceding paper, that when any complete solution 3(x) has been found, it is 
possible to derive from it a general solution with one disposable constant C, 
,-K«- i)“l 
Accordingly, it is convenient to deal only with the 
C3 
xG 
dt 
particular solution for which t = 1 ; ^ = 0 ; at a? = gc , denoted by ® K (x), and 
called the Homer Lane Function. 
§ 3. Homer Lane, in his paper “ On the theoretical temperature of the 
sun,” gives analytical solutions of equation (1) for the cases k — If and 
k — L4, which correspond to a monatomic and to a diatomic gas respectively. 
His method of obtaining these solutions is equivalent to the following. 
Assume 
t = ®(x) = l+“i + “ 2 
' x 2 x; 
4+~6 + 
etc. 
( 2 ) 
to be the required solution of (1), where cq, a 2 , a 3 , etc. are to be determined. 
Then 
<m K 
dx 2 
2.3.a n 4.5.a 0 6.7.a, 
+ etc. 
(3). 
And the coefficients cq, a 2 , cq, etc. can be determined from the equation — 
2.3.cq 4.5.a 2 6. 7. a. 
4 -o— + etc. 
CL~t CL<y Otq 
1 4 — oH — > 4 — h + etc. 
ry*£ n/ »*± /y»0 
tAy tAs 
The solutions given by Homer Lane are, in the present notation, 
1 1 
®1 -o( X ) — 1 - + ~Q~r\ i 
6x 2 8 Ox 4 1440x 6 31104x 8 
® 2 . 5 (x) == 1 - t^o + 1 
125 
6x 2 48x 4 201 6x 6 435456x 8 
.... etc. 
. etc. 
(4) 
(5) ; 
( 6 ) . 
These terms are sufficient to give a satisfactory determination of t for all 
values of x equal to or greater than unity ; that is, at all points in the gas 
within distance cr from the centre of the sphere, where the radius of the 
boundary is cr/q, and q is the value of x for which © (a;) = 0. 
The labour of calculating additional terms of these series being very 
great, and no great precision being necessary, Homer Lane merely employed 
a step-by-step process, involving the use of numerical differences, to obtain 
approximately the value of t at points whose distance from the centre of 
the sphere was greater than that corresponding to x unity in equations (5) 
and (6). When a fairly small value of t had been reached by this method, 
he was able to complete the calculation as far as t = 0 (x — q) by means of 
approximate formulas which can be derived in a manner similar to that 
described above. 
