394 Sir William Thomson on the General Integration 



Of these equations, (29), (31), the second of (32), the second 

 of (33), and every second equation thenceforward, determine suc- 

 cessively Kj, K 3 , K 5 , K 7 , and so forth, all in terms of the arbi- 

 trary C; and (30), the first of (32), the first and third of (33), 

 determine successively K 2 ,K 4 ,K 6 , &c. in terms of the arbitrary K . 



7. Returning now to the more general supposition of the 

 depth varying with the latitude, we may assume, without prac- 

 tically restricting the problem further, 



7 = 7o + 7# + 72/^+---+y,X; • • (34) 

 7o) Yv ' • • Vn heing given constants. This makes 



7(*)K i =*7 K|+ttK«'i 1 +y 2 K *-2 + •..-• +7^_ n . • (35) 



Using this in (25) and proceeding precisely as in § 6, we find 

 Kj, K 2 , K 3 , K 4 , K 6 , &c, each in terms of two arbitrages C and 

 K — unless 7 contains only even powers of jjl, in which case, as 

 in that of uniform depth (§ 6), we find K T , K 3 , K 6 , ... in terms 

 of one arbitrary C alone, and K 2 , K 4 , K 6 , ... in terms of the 

 other arbitrary K alone. The first two of the equations by 

 which this is done, those namely which correspond to (29) and 

 (30), being found by putting i = and i = l in (25), are 



K,-(^-^)c = (36) 



and 



7o V / 70 7o' / 2 7o 



8. In §§ 5, 6, 7 we supposed <£> = 0, and so made, for the 

 time, "free oscillations" our subject. Now suppose <I> to be 

 any given function of /jl. For the actual problem of tides of 

 any species, it is a rational integral function of //,, or of jj, and 

 V(l —^), if we neglect the influence produced by the change of 

 attraction of the water due to its change of figure. A proper 

 way of taking into account this influence by successive approxi- 

 mations will be explained later. Meantime, without losing gene- 

 rality, I assume 



<S> = «£> + <S> l/ 4 + <S> 9 ^ 8 + . . . + <V + &c, . (38) 



where <I> , <£„ <E> 2 > &c are given constants, either finite in num- 

 ber, or of such magnitudes as to render the series convergent for 

 values of yt, within the limits used in each particular case. With 

 this for C X>, the second member of (14) becomes 



-4mV(^~^- 2 ), (39) 



and instead of (25) we have 



