of Laplace's Differential Equation of the Tides, 391 



d$ 



1 / ' /3N-7+1 dfJb 



1 # 



°-4m^ U ^ 'l/*^ /W0J j 



(13) 



(!-/*«)« 



Using these instead of (8) in the process by which (9) was found 



above, and multiplying the resulting equation by 4<m (sin 0)7 + *, 



we find 



d 7 # . 9 fs , ^7/^(1-^) # 

 dpf*-M*dp + *V V P-^ ifi 



+ [-f 2 7 + 4^(l-^ 2 )]^=-4m(l- A 6 2 )^. (14) 



5. To integrate this, take first the case of $ = (free oscilla- 

 tions), and assume 



1 



d<b 



= K + K # + K 2 ^-f . . . +K,^ + &c. (15) 



This gives 



£=C+/%^ + f / 2 K l/t *+ f (/ 2 K 2 -K ) + . . . 



+ ^(/^K i _ 1 -K i _ 3 )+&c. ) . (16) 



where C denotes a constant of integration. Now let -ct denote a 

 symbol of operation such that 



otK^K^m or generally wF(i) = F(i-l), . (17) 



F(z) being any function of z". By aid of this notation we may 

 write (16) short thus, 



^=V!(/ ! -^)^,.; . . .. . (18) 



understanding that, when z = 0, 



/ 2 K / _ 1 — K f _ 3 = ^ 



aud that 



K { =0 for all negative values of i. 



(19) 



(20) 



Let now 7 (-or) denote a symbol of operation obtained by put- 

 ting ts- for ft in y (which, be it remembered, is a function of //<). 



