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 1 + BP 3 + GP 5 (where A, B, G 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 1 its expansion. Then (43) becomes 



( f i-b ) + 3(a 1 -aV) 



|+|p 2 + &c.4-(-l) 1+m ' 2 x 



M:::<:+S (a " +1)P - +ta 



= -2,(2n+l)a n P n (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 



A*-&o + !(or-o7)«0 (45), 



3 



-- 5 (a 1 -aV) = a 2 (46), 



an( 



3(a 1 -aF)(-l)^x|^-^ T | ) ) = a w (47), 



in the last equation n being supposed 4 or some higher even 

 number. 



From (35), (45), and (46) we shall get 

 ^i = -^ + a +\aV-^(fM-b )l^P 1 + 



+ ^-^P a + S(^„)..,(48). 



If y\r be any stream-function, and <jb its corresponding velocity- 

 potential, it can be shewn (Rayleigh on Sound, Art. 238) that yjr and 

 (j> are connected by the relations 



%-%*" ^ t=t^ e < 49 > 



Putting here for (f> the value of <p x from (48) and taking account of 

 the equation which P n satisfies, viz. 



d^[ . n dP. 

 dd 



(sme^) + n(n + l)P n sm6 = (50), 



