278 On a controverted Point in Laplace's Theory of the Tides. 



8(K 2 -G) =0; 

 12(K 4 -K 4 ) = 0; 



(-16K 6 +10K 4 )^-4K 2 =0; 

 (_40K 8 + 28K 6 )^--4K 4 =0; 

 and generally, after the first two, 



Now an equation like this, which gives the third term in terms 

 of the first and second, the fourth term in terms of the second 

 and third, and so on, must have the first and second terms as 

 foundations or origins wherefrom to start for the formation of 

 the entire series. There is no more ground for inferring the 

 second term from the general subordinate equation than for in- 

 ferring the first term from it. The first and second terms must 

 stand on their own foundation. The value of K 2 is perfectly 

 clear, namely K 2 = G (not deduced from the general subordinate 

 equation). But how with regard to K 4 ? The subordinate 

 equation 12(K 4 — K 4 ) = gives no information whatever; and I 

 infer from this that we may give to K 4 any value that we think 

 fit (as regards the equation, unlimited by boundaries &c, with 

 which we started), or any value that may be required by physical 

 circumstances not yet introduced (as the boundary of the sea). 

 But does any arbitrary value of K 4 really satisfy the original 

 differential equation ? It does. Then I say that K 4 is truly an 

 undetermined quantity, which may perhaps be determined here- 

 after by reference to physical conditions not yet stated, but 

 which cannot under any circumstances be inferred from the general 

 subordinate equation. And, so far as I know, the whole course 

 of applied algebra does not offer a more certain conclusion. I 

 scarcely need to add that I look upon Laplace's process as a 

 mere sport with symbols, and upon Laplace's conclusion as a 

 grievous error. 



In Article 112 of my Essay, I have given the series which 

 ought to be added to Laplace's series in order to make the solu- 

 tion complete; and I now repeat it as accurate (chance-errors 

 of numbers excepted). The additional series is the following : — 



K 4 x{si„^ + |siu^ + (l-^)siu^ 



, /189 79 bm\ . ... . T 

 + (576-730-> m '°* + &c -j' 



