556 VISCOSITY. [CHAP, xi 



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

 denned by 



1.3...(2re + l) 2(2 + 3) 



...... (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 



am' + yv f + zw f = ........................ (5). 



The typical solution of the Second Class is 



t' = (n + 1) ^ M (Ar) ^P - mK +1 (ArJ 



M/t</ /""" ' - 1 



d <f> n / a \ 



j--^>i'-W> 



-*~ ^ l+1 dz 



where fyn is a solid harmonic of positive degree n. The coefficients 

 of tyn-i (hr) and ty n +i (hr) in 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 



*.'(> >.H..(f) (7), 



r*.' (0+(2+i)*. (-*._. (0 (8), 



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



asu' + yv' + W = n (n + 1) (2w + 1) ^ (Ar) ^ (9), 



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

 If we write 



dw' dv' 9 ,_du' dw' dv' du' 



28 = 5- r~ > ^ "^ T~ i -"b == ~? T~" 



6 ^2/ dz dz dx ' c?a? dy 



