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. 357
which may also be written
8k20r2 f $(dI + (dd)2 + (rD )2 - Y 2 ( 2
87vrkor (2 \ dx \dy \ dz - 2
+ dy/ + d2 - 2 dS.........(58).
The expression under the integral signs in (58) is equal to the sum of the squares
of the tangential components of the vectors
(dn/dcx, d),,/dy, d,,l/dz) and (dbn,,/dx, d,,/dy, dc/,dz).
Now if S, be a surface-harmonic of order n, we have
r27r & \2 d 7. 2rn
f ( d^) + ( d-Y ) } sin 1OdOdw = n ( + 1)j SS,' sin Oddoo......(59)*.
Hence (58) may be written
n ( + 1) 8 2 2n+ ( +,,2) dS, or n (n + 1) 8. 2 dv............(60),
where IT,,' denotes the modulus of T,', and d-r is an elementary solid angle, viz.
dS = r2dz.
In a similar manner, a solution of the second type gives the result
(n + 1) 2d.................................(61).
It appears, further, on examination, that the parts of the expression (53) which
arise from combinations of the two types, or from combinations of the same type with
different values of n, will disappear in virtue of the conjugate property of surface-harmonies
of different orders.
Hence, if Z be a sign of summation with respect to n, the general expression for
the rate at which energy is dissipated by the scattered waves is
8 n (: n ) { | T,1 + | U,,,2} d-,..............................
In the case of plane incident waves the harmonies are tesseral, of rank 1. Writing,
for shortness,
Tn' = BnT^, U=.................................... (63),
* Proved easily by partial integration, making use of This involves products of surface-harmonics of orders
the differential equation m -1 and n, and will therefore vanish unless m= n + 1.
1 _d.dS N 1 d2Sn But writing it in the form
d si + Ls i+ - n+(12 +.1) OS, - 0.
sin 0dc dO )sin20 dX2 (+olX z 15?
f The integrals which arise from combinations of the J x \ dy dz
two types are of the form we see that it also vanishes unless n=Lm+l. Hence it
II [ x {......{ vanishes i1n any case.
J jn (II d lx dz dXn'y _ \