234 Mr Sharpe, Liquid Motion from a Single Source 
Legendre coefficient of odd order can be expanded in a series of 
the same coefficients of even order, and of course any expression of 
the form (say) AP X + BP S -f CP 5 (where A, B, C are arbitrary) 
could be similarly expanded. Applying this to (43) and remem- 
bering that the constants a n for n = 2 upwards are disposable, it is 
easy to see how (43) can be satisfied. 
Probably the simplest case is the following. In (43) put for 
P x its expansion. Then (43) becomes 
(p - b 0 ) + 3 (oj - aV) 
| + |p 2 + & c. + (- iy+«x 
X 
3.5 ...(w- 3) 
2 . 4 . . . (n + 2) 
(2 n+ l)P n + &c. 
00 
= - 2 (2 n+ 1) a n P n 
2 
(44). 
On both sides n is now supposed even, but on the left-hand side 
the least value is supposed to be 4. (44) is satisfied if 
P -6° + ^{cby - aV) = 0 
(45), 
-~(a 1 -aV) = a 2 
(46), 
and 3 (a, -aV){- l)’*/ 3 x 2p’’(,” + 2 ) = a, ‘ ( 47 )> 
in the last equation n being supposed 4 or some higher even 
number. 
From (35), (45), and (46) we shall get 
0i = 
-^ + a , + \aV-\{ l ,-bA r -P 1 + 
r 2 no 
a„r 
,) ...(48). 
If ^ be any stream -function, and </> its corresponding velocity- 
potential, it can be shewn (Rayleigh on Sound , Art. 238) that fa and 
< p are connected by the relations 
= — C -j i sin 0 and C ~ = ™ r 2 sin 6 (49). 
dr dd dd dr 
Putting here for </> the value of fa from (48) and taking account of 
the equation which P n satisfies, viz. 
^sin 6 +n(n + 1) P n sin 6 = 0 
(50), 
