(40) 



of Laplace's Differential Equation of the Tides. 395 

 |[i(l-^ 2 )+l + (j+l)^](l-^)7H 



The proper modification, according to this formula, must be 

 made in (27), and in each of the particular equations (29), (30), 

 (3J), (32), (33), (36), (37) when required. 



9. Before considering the conditions which may be fulfilled 

 by proper determination of the two arbitrary constants C and K , 

 it is convenient to investigate the convergency of the series (16) 

 which we have found for the complete solution. For this pur- 

 pose put (40) [including (25) as the case for which <^ = 0] into 

 the form 



(l-m 2 ) 2 yWK i+1 =i{-[l + (j + l)^](l-^) 7 WK i+1 



+ |^7«-4«n(l-^]i(/ 4 K,- 1 '-Ki-3)-4«(*i-*i_ a )J. (41) 



In a certain very important class of cases, of which the first 

 example known to mathematicians is that so splendidly and suc- 

 cessfully treated by Laplace in the process defended and contro- 

 verted in the two preceding Numbers of this Magazine, terms of 

 the second member of this equation are, for infinitely great 

 values of i, comparable in magnitude with terms of the first 



TC- X- 



member, through ■ T J~ 1 or „ * , being infinitely great of the 



order i 2 . These cases can only occur when y is either constant 

 or expressed in (34) by the first two terms, ry -f- «y 1 yt6. Reserving 

 them for consideration later, we see by (41) that, except in those 

 special cases, K* must for very great values of i fulfil, more and 

 more nearly the greater is i, the equation 



(1-«*)M«OK<+,=0 (42) 



Calling Ki the complete and rigorous solution of this equation in 

 finite differences, we have 



Ki = l + h+Q» + l»>i)(-iy+t i+ V i + s ZCt) . . (43) 



r P 



where p } p', &c. denote the roots of the equation y = 0, and /, V, 

 l", l ! ", k, k', &c. constants. Hence for great values of i, K t must 

 be approximately equal to (43) with some particular values for 

 the constants /, /', &c. But for very great values of i all the terms 

 of (43) except one leading term, or [because of the equal roots of 

 (1 — w 2 ) 2 = 0] one leading pair of terms, vanish in comparison 



