excited Vibrations of an AtmospJiere. 153 
us write 
R.H. member of (23) = Sj cosfe+S/ sin kt 
+ (So + S, + S G +...)cos2/^ + rSo / + S 1 ' + S G '...)sm2^ 
+ • (24) 
where eacli S is a surface harmonic (of degree indicated by 
the suffix) multiplied by a constant physical factor. For the 
determination of the disturbance of pressure Bp, we may 
assume provisionally a solution of (17) in the form 
&P= Qi cos kt + Qx' sin kt + (Q 2 + Q 4 + Q 6 + • • ■) cos 2kt 
+ (Q 2 ' + Q,' + Q 6 / +...)±m2Lt + ...; (25) 
the Q's being also surface harmonics multiplied by constant 
physical factors. Now 
O?' >' o>' rsm0 a#\ 00/ ?-snr0 o<o- 
and if ^ is the outward radial velocity of the air at any point 
within the spherical sheet, we have from (5j , taking the axis 
(say) of z. vertically upward, 
p-=^i = ^7 (p being independent of t). . (27) 
But q vanishes at both spherical boundaries of the sheet, 
the radii of which may be called r and r + Ar : hence when 
Ar is very small, q is evidently insignificant throughout the 
motion; so that 
!?=0, and ^=0 (28) 
Ot or 
Also 0=(q at outer boundary) — (q at inner boundary) 
= oq/orAr, so that oq/or = always. 
Hence by (27) yfy/to*=0, (29) 
which justifies our assumption that Bp is independent of r. 
Using (28) and (29), (26) becomes 
V'%= i | J-. * (sin 0*)+ J-, ^jlQiCos fc + Q/rin kt 
1 r* Lsin0o0\ 00/ snr o© J 
+ (Q 2 + Q i +Q6+...)cos2^ + (Q 2 , -hQ/ + Q 6 ' + ...)sin2B} 
= - y (Qi cos ** + Qi' siu /j ^) - y (Qa cos 2/ ^ + 0a' sin 2 *0 
-^r (Q* cos 2kt + Q/ sin 2ft) - ^ (Q 6 cos 2fe + Q 6 ' sin 2kt) 
-,... , (30) 
by a well known property of surface harmonics. 
