ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 155 



We may also write these equations in a variety of alternative forms in which 

 prominence is given to any one we please of the quantities M', N,. These forms 

 are obtained by giving n different integral values in the equations 



= 0. 



NS |"o.'-i '- 2 <. "I r ; ;+i .+ "1 n 



*~ M - N" - M' AT" V M' 



L Ji -l iN n-S ""i-a J L lU /( + l ^n+i iu +:: -J 



In each case the former continued fraction terminates with a partial quotient 

 involving a", in the numerator and either M^ or N* in the denominator, while the 

 latter proceeds to infinity. 



For the symmetrical types, if we use the form (56) we must suppose n .t an 

 even integer, whereas if we employ (57) n * must be supposed odd. The reverse 

 will of course be the case for the asymmetrical types. 



The continued fractions of the present section will not converge so rapidly as those 

 of the preceding, but in spite of this drawback they present considerable advan- 

 tages. In the first place the numerators of the partial quotients, which are obtained 

 by giving n different integral values in the expression (54), are independent of X. 

 These numerators, which further are in a convenient form for logarithmic computation, 

 may therefore be tabulated once for all, whereas the numerators of the partial 

 quotients in the continued fractions of the last section require to be re-determined at 

 each successive trial in attempting to solve the period-equation by trial and error. 

 Moreover, the evaluation of the denominators M',, N;\ by means of the formulae (51) 

 may be very quickly effected, even though a fairly large number of these denomi- 

 nators is required, whereas the evaluation of the quantities L* by means of (29) and 

 (32) is extremely laborious. 



Another disadvantage resulting from the use of the preceding form is that when 

 X/to is near the value 2s/n(n + 1) the functions x'_ lt yl^ both become large, while 

 Ll-b L+i have both a zero and an infinity in the immediate proximity of this 

 value. Hence, in order to evaluate L^, L* l+1 for a value of X in this region it is 

 necessary to observe a very high degree of accuracy in the numerical work. The 

 singularities which occur in the left-hand member of (46) when X passes through one 

 of these critical values no longer appear if we write the period -equation in the form 

 (57) with the value of n appropriately chosen. 



7. Expressions for the Velocity -Components. 



The auxiliary constants Dj, Dj +1 , &c., introduced in the last section, may be made 

 use of to express the velocity -components by means of series of surface-harmonics. 



X 2 



