556 VISCOSITY. [CHAP, xi 



where ^ n is a solid harmonic of positive degree n, and ty n is 

 defined by 



d l sin 



1.3...(2 + 1) 



(4). 



It is immediately verified, on reference to Arts. 266, 267, that the 

 above expressions do in fact satisfy (1) and (2). It is to be 

 noticed that this solution makes 



xu + yv + zw f = (5). 



The typical solution of the Second Class is 



u = (n + l) ^ M (hr) --p - n^ n+1 (hr) h 2 r +B - r 



dx 



J fk 



(hr] 



where &amp;lt;j&amp;gt; n is a solid harmonic of positive degree n. The coefficients 

 of ^ n -i (hr) and ^r w +i (^) i n these expressions are solid harmonics 

 of degrees n 1 and n + 1 respectively, so that the equations (1) 

 are satisfied. 



To verify that (2) is also satisfied we need the relations 



* (0 = -Ww.&amp;lt;0 ..................... (1), 



which follow easily from (4). The formulae (6) make 



xu + yv + zw = n (n -f 1) (2w + 1) ^r n (hr) $ n ...... (9), 



the reduction being effected by means of (7) and (8). 

 If we write 



dw dv ,du dw dv f du 



