ANALYSTS TO THE DYNAMICAL THEORY OF THE TIDES. 157 



(n + s + 1) 



The formulae (61) cease to be of use in a special case which will present itself here- 

 after for which K = 0. It will be seen in a later section that the expressions on the 

 right become indeterminate in this case, so that the determination of the velocity- 

 components must be effected by means of the formulae (12) or (13). These latter 

 formulae seem at first sight to indicate that the velocity-components become infinite 

 in latitude sin" 1 ($/cr), but the forms (61) indicate that such cannot be the case, at 

 least when K is different from zero. 



8. Approximate. Determination of the Higher Roots of the Period- Equation. 



If we take the period-equation in the form (56), and as a first approximation omit 

 the continued fractions from the left-hand member, it reduces to the quadratic 



M:, = o, 



or 



For large values of n the roots of this equation will give a sufficiently accurate 

 approximation to the roots of the period-equation, since it may be seen that the 

 continued fractions tend to limits comparable with Jr/i 3 , and therefore small in com- 



parison with n a (n + I) 3 " a , when n is very large and X/&> has as its value either of 



~X.G) Or 



the roots of this equation. We may even obtain a fair approximation by omitting 

 the term containing s, in which case the formula for X corresponds with that obtained 

 when the rotation is omitted. 



A better approximation will however be obtained by representing the continued 

 fractions by their first convergents instead of entirely neglecting them. The 

 approximate form of the period-equation is then 



or 



_X^ fo(m+l) 2(as/\} (n - I) 2 (n - s) (n + s) 



4w 3 " n z (n + I) 2 " n 2 (2n - 1) (2n + 1) {(n - l)n - 2, 



(n + 2) 3 ( - s + 1) (M + s + 1) 







" (n + I) 2 (2ra + l)(2t + 3) {(71 +"l) (n + 2) - 2a>s/\} 4w 3 ffl 2 ~ 



We thus get back to the equation Lf, = 0. The roots of this equation may be 

 approximated to numerically by HORNEE'S process, the significant roots being those 

 which lie in the intervals between oo and 0, and between 2s/n (n 1) and + 



