in Laplace's Theory of the Tides. 231 



" escaped Laplace, and he has actually persuaded himself to 

 "adopt the following process. Putting the general equation 

 (t among the coefficients into the form 



o hm 

 ^2k+2 __ / 



K 2* TTTTT^ ZZ . „„ Kaft+4' 



{2k <2 + 3k)-{2k 2 + 6k)~ 



2k + 2 



^4 = 



K 2 



" he has unwarrantably conceived that this must apply when k = l 

 u for the determination of K 4 ; and thus applying the same equa- 

 " tion to each quotient of terms which occurs in the denominator 

 u of the fraction, lie finds 



9 bm 



2.1 2 + 3.1-(2.r 2 + 6.1)x L 



2.2 2 + 3.2-(2.2 2 + 6.2)x 



bm 



2.3 2 + 3.3-&c. 



li in an infinite continued fraction. And upon this he founds 

 " some numerical calculations adapted to different suppositions of 

 " the depth of the sea. We state, as a thing upon which no person 

 " after examination can have any doubt, that this operation is en- 

 tirely unfounded/' 



7. A careful examination at the time when I first read this 

 led me to the opposite conclusion, and showed me that Laplace 



was perfectly right. If * k+2 vanishes when k is infinitely 



great, then ~ cannot but be equal to the continued fraction. 



2 



What, then, must be the case if K 4 has any other value than that 



determined by Laplace? -! /c+2 cannot then converge to zero 



for greater and greater values of k. But unless J^"" 2 is infi- 

 ll 2k 

 nitely small when k h infinitely great^ the second term of the 

 second member of (6) is infinitely small in comparison with the 

 first, and therefore ultimately 



„ _ 2k + 3 v 

 t+4 ~ 2k + 6 ' 



