of Edinburgh, Session 1871-72. 
679 
Hence 
W = i^Ar(P + +)f, . . 
(32). 
The condition that with velocity-potential P -J- 4* there is 
dicular to the spherical surface, gives 
no flow perpen- 
O 
II 
e 
II 
Si 
+ 
• (33). 
Now let 
P = P « + P .a+ + Pi G)‘ +&C ' 
+ = *((f+ + *G) , + 1 + ta. 
| • (34), 
be the spherical harmonic developments of P and vf, relatively to the centre 
of the rigid globe as origin, the former necessarily convergent throughout the 
largest spherical space which can be described from this point as centre 
without enclosing any part of the core ; the latter necessarily convergent 
throughout space external to the sphere. By (33) we have 
= Pi 
* + l 
(35). 
Hence (32) gives 
which, by 
becomes 
w =#K^i p 0( aP -)’ 
jrdaVft = 0 , 
( 86 ). 
Now, remarking that a solid spherical harmonic of the first degree may be 
any linear function of x, y , z, put 
which gives 
and 
1 
P^ = Acc + Br+C? 
£ 2 = A 2 + B 2 + C 2 , 
(37), 
~ JJ fcPi = (A 2 + P 2 + C 2 ) . | .Jfdtr = gT X volume of globe = ? pq * . 
Hence by (36 
W = J + + +...) . (38); 
and, therefore, by comparison with (31), 
2.5 , 3 . 7 T 
( 39 ), 
