230 Sir William Thomson on an Alleged Error 



fied by 



that is to say, agreement with the equilibrium tide. 

 V. Lastly, Laplace and Airy assume 



^=K^ + K 4 ^ 4 + ... + K 2 ^+K 2 , +2 ^ +2 + &c. . . (3) 



as a solution of the differential equation (2) . Then, by equa- 

 ting the coefficient of x 2, in the left-hand member to — 8H, its 

 coefficient in the right, and by equating to zero the coefficient 

 of cc 2k+i for all values of k from to oo, they find 



-2.4K 2 =-8H (4) 



and 



2k(2Jc + 6)K 2k+4 -2k(2k + S)K 2k+2 + 4n ^ K 2 ,=0 . (5) 



r 



for all positive integral values of h [the case of k = justifies 

 the omission of K in (3)]. The first of these equations of 

 condition gives K 2 =H. The second, if for brevity we put 



mr 



— = e, S l ves, 



^ 2k+ *=2kTS K2k+2 ~W^) K2lc) ' * ' (6) 



and so determines successively K e , K 8 , K 10 , . . . &c, all in terms 

 of K 2 , K 4 . Thus the differential equation (2) is satisfied by (3) 

 with K 2 given by (6), K 4 arbitrary, and the other coefficients 

 given by (6). 



6. On this and Laplace's process for completing the solution 

 Airy [art. (Ill)] remarks : — 



"The indeterrninatenessof K 4 is a circumstance that admits of 

 " very easy interpretation. It is one of the arbitrary constants in 

 " a complete solution of the equation. It shows that we may give 

 " to K 4 any value that we please, even if H* = 0; and then, pro- 

 " vided that we accompany our arbitrary K 4 with the correspond- 

 " ing values of K 6 , K 8 , &c, we shall have a series which expresses 

 " a value of a f that will satisfy the equation when there is no ex- 

 <{ ternal disturbing force whatever, and which therefore may be 

 " added, multiplied by any number, to the expression determined 

 " as corresponding to a given force. In the next section we shall 

 " find several instances exactly similar to this. Yet this obvious 

 " view of the interpretation of this circumstance appears to have 



2L 



* G in Airv's notation, in Laplace's. 



t ga in Airy's notation, oca in Laplace's. 



