Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 351
When values of X, Y, Z satisfying these equations have been found, the corresponding values of a, /3, y are given by (10). Or, we may reverse the procedure,
determining the general values of a, f, 7 by means of equations similar to (11) and
(12) and thence the values of X, Y, Z by means of (9).
The solutions of (11) and (12) subject to the condition of finiteness at the origin
are of two types. In the first place we may have
X = {hrh,î' n(h + )) +(h (n + 1),nhr (hrr) xr-2 T,
Y= {hrqfJ' (hr) + (n + 1) Jr (hr)} rT - lTn~rq' (hr) yrn-2Tn, -...... (14),
dy
Z = {[hrn (hr) + (n + 1) % (hr)} d rTl nhr i Jr' (hr) zr"-2 T
where Tn is a spherical surface-harmonic of order n*. These make
io-K (ci /d d m
=-iK I (hr) (y -z ),
ciK dx d\
=-iE-, (hr) z- dx -- )rl2^T TT. l)j
( dx
=-K C h (hr) d-nT
It follows that
xX + y Y + zZ = n (n + 1) (hr) n...........................(16),
xcO + y + zy =0.(;............................................. (17);
also that
yZ - zY = hr,,' (hr) + (n + 1) (hr)} (y -z r', &c., &c............(18),
jo-K __ nxrd~ 2T~
y-z3fi= = r2r, ((hr) r1T (d -nxr' -2T,, &c., &c............................9)
In the solutions of the second type we have
a = {hrn' (hr) + (n + 1) (hr)} - r U - nhrf,' (hr) xr'2U
a = ihrefrn' (hr) + (n + 1) Jn (hr)} fdr' U - nhreJt' (hr) yr2 Un,.........(20),
dy
y = {hrJr,,' (hr) + (n + 1) ^ (/tr)} - rUn - nhrl4t,' (hr) zrV-2Un
* These are equivalent to the forms given iin Hydro- formula relating to spherical solid harmonies, such as
dynamics, ~ 305 (6), divided by 2'n+1. The proof of the =r2 (ddp n,+1 \ d n )
equivalence requires the use of (6) and (7), together with '=t 2?+1 \ dx dx r2n~1) *