of Laplace's Differential Equation of the Tides. 401 



Then, putting 



p=z+a (60) 



in § 4 (14), and working precisely as in § 5, but with z instead 

 of /x in the second member of (15) and the proper corresponding 

 modification of (16) &c, we obtain a solution in ascending 

 powers of z, or ft — a, which is necessarily convergent through- 

 out the range of our problem. The degree of convergence of 

 the series so found for each of the six functions, 



«{z + a), 0(*+«), \{z + a), 



*{z + a), p{z + a), X(^ + fl)> 

 is, for any value of z, the same as that of the geometrical series 



±:+G> 



where c is the less of the two quantities a—p, p 1 — a. 



16. For our second proposed case (§ 11) let p p p, p', p !r be 

 four consecutive roots of (1 — ^ 2 )y = 0; let p } p' be each between 

 — 1 and +1; and let 7 be positive for values of p, between p 

 and p 1 . Required, to determine tides, and the free oscillations, 

 of the zone of water corresponding to these intermediate values 

 of p. Take any quantity, a, between p { and p n , such that p ~ a 

 and p' ~ a are each less than the less of the two differences a—pp 

 p" — a. Put p, = z + a, and solve in ascending powers of z, as in 

 § 15. Let ot i} yS/, \ be the coefficients of 2? in the series thus 

 found for a(p,), /3(fi), \(fi) in formulae corresponding to (50), 

 but with z for p, in the second members, so that we have 



/■z^5=CW)C+G5A^K +G6yOL. . (61) 



Let now p, q be two values of i, and put 



«/!+&K +\,L = 0n 



«C+&K +X g L=0.J [ } 



If p, g, and p — q* be each infinitely great, the values of C and 

 K determined by these equations and used in (61) and (55), give 

 the tides due to the tide-generating influence 



<& = L(n + n 1 p, + n 2 fj,' 2 + &c). 



The periods of the fundamental free oscillations of the supposed 

 zone of water are determined by finding <r so as to make 



£-% (G3) 



* Except mthe ease of p — a = — {p' — a), when we must take;; — q = l, 

 or any odd integer; hut p — q = \ is best in this case. 



Phil, Mac/. S. 4. Vol. 50. No. 33.2, Nov, 1875. 2 D 



