of Laplace's Differential Equation of the Tides, 393 

 and expanding the first member of (25), we have 



(* + l)K l+1 +(-2 J -+^-^- 2 )k,. i 



♦[^-■■-Tf-SK«+H^ (B > 



For all negative values of i up to —2 this equation is an identity 

 in virtue of (20) ; and for i— — 1 it becomes 



0K =0, ....... (28) 



and leaves K arbitrary. For 8=0 it becomes, in virtue of (19) 

 and (20), s^-af* 



K.-Ay^-C-Oj (29) 



for f=lj in virtue of (20), 



2K 2 +(-2+^- S 2 + a /' 2 )K =0; . (30) 

 and for 2 = 2, in virtue of (19), 



3K 3 + (-4+??-^^)k,-«C = 0. . . (31) 



Then directly, for z'=3, 2 = 4, 



4K 4 + (-6 + ^-^) 



+(2-7 + ^-«/ 2 )k =0, 



KIT , 7 a , 2s , * 2 -»/ 2 \t^ 

 6K 6+ (-10+i s -^^)K 4 



, / . 2S S 2 -«/ 2 «/*\ T , «, 



+ ( 4 -7 + -5F--i) K ^3 K o= ^ 

 7K 7 + (_12 + ^-^)k 5 



+ \ 7 + "e/^ ~ 17 Ka+ 4 K ' =0 ' 



8 K8 + (-14 + ^-^),K 6 



,(32) 



J- (83) 



and so on. 



+ (e-.| + £W!-ff)K, + i K„o, 



