293-294J SOLUTION IN SPHERICAL HARMONICS. 529 



Hence the equations (1) are satisfied, provided 



Also, substituting in (2), we find 



l 



whence B=, - ... /rt - rrr^ - ;rr~ ................. CO- 



Hence the general solution of the system (1) and (2) is 



u = 



* dpn 



2(2n + l) dx 



= lvf __ rL 



H (2 (2n + 



, - a Pn , , 



(n + 1) (2n + 1) (2n + 3) dy r+^ 



^ \+w 



1) efe (n + 1) (2?i + 1) (2w + 3) dz 



......... (8)*, 



where w , v 7 , w have the forms given in (9) of the preceding Art. 



The formulae (8) make 



nr 2 



(9). 



Also, if we denote by f, 77, f the components of the angular 

 velocity of the fluid (Art. 31), we find 



dz dy J dx 



o ^ V -^ f ( * / Pn dp n \ ^ / ,T\ ( * Xn \ /in\ 



Zri = Z ; , . /rt 77 z -j- x -j- I + ^- (?i + I ) ~= , V...( 1U). 



/A (n + 1) (2r? + 1) V # dz J dy 



These make 2 (#f + 3/17 + ^f) = Sn (w + 1) XH (H). 



* This investigation is derived, with some modifications, from various sources. 

 Cf. Thomson and Tait, Natural Philosophy, A.rt. 736 ; Borchardt, 1. c. ; Oberbeck, 

 &quot;Ueber stationare Fliissigkeitsbewegungen mit Beriicksichtigung der inneren 

 Reibung,&quot; Crelle, t. Ixxxi., p. 62 (1876). 



L. 34 



