First Species" in Laplace's Theory of the Tides, 281 



[which substantially agrees with Airy's equation of art. (97) 

 (with <7 = 0, to make the depth constant as we now suppose it), 

 and with Ferrel's § 151, equation (288) ; but is simpler in form, 

 partly through the use of Laplace's notation jju for cos 6] . For 

 each of the " long-period tides " in the actual case of the earth 

 under the influence of the sun and moon, the function © is given 

 by the formula 



e»H(i-v>, (3) 



where H denotes the equilibrium value of the tide-height at the 

 equator. Airy, with this value of ®, finds an integral of the 

 differential equation by assuming 



a' = B 2 //, 2 + C 4/ u 4 +...B + &c, 



and determining the coefficients so as to satisfy it. But this 

 assumption errs in making the tide-height at the equator equal 

 to the equilibrium height. The correct assumption for the par- 

 ticular problem proposed (or for any case in which (8) involves 

 only even powers of //,) is 



a i = B + B^ + B 4 ^ + ...; 



but the more general assumption, 



«' = B + B^ + B 2 ^ 2 +... +B./^-f..., . . (4) 



is as easily dealt with (and includes oscillations in which the 

 equator is a line of nodes) . With it we have 



-(i+2)(i+l)B i+2 -4eB i }, 

 which is to be equated to 4e0. Thus, for the case of 



© = H(1-3 A 6 2 ), 

 we find, by putting i= —2, i=0, 2 = 2, &c. : — 



2.(-l).B 2 =0, ■> 



4.1.B 4 -2.1.B 9 -4eB =4eH, L . . (5) 



6.3. B 6 -4 . 3 . B 4 -4eB 2 = -12eH, J 



and 



(f + 4) (t + l)B t - +4 - (i + 2) (* + l)B l+2 -4 e B f =0, . (6) 



for all even positive values of i except and 2. 



The first of equations (5) gives B 2 = 0; and with this the 

 second and third give 



B 4 -«(B + H),\ 



B e =f e B ; J (?) 



and if in (6) we put successively i=4, i=6, j = 8 ; . . . and use 



