178 MR. S. S. HOUGH ON THE APPLICATION OP HARMONIC 



tide at the equator is 3(8%, and therefore the ratio of the height of the tide to that 

 of the corresponding equilibrium tide at the equator is 



+ 7-9548. 



In like manner the tide-height in any other latitude may be compared with the 

 equilibrium tide-height, but the process will be laborious in the absence of tables of 

 the functions P*.* 



The above example has been treated in some detail as illustrative of the method to 

 be employed for the computation of the forced tides by infinite series ; the chief part 

 of the labour is involved in the determination of the quantities af M i/',, A',, but when 

 once these have been determined, since they do not involve the depth, it is easy 

 without much additional labour to multiply cases for different depths. Besides the 

 case already considered, which corresponds to a depth of about 7260 feet, I have 

 computed the series for depths of 14,520, 29,040, and 58,080 feet, corresponding 

 with the values -^j, -($, and ^ for hg/ara~. For these depths the series within the 

 square brackets is replaced by 



- 0-83227P; + 0-21G94P; - 0'02G15P; + O'OOlSOPj; - 0'000080P? 



+ -000003 P? 2 + ..., 



- 191'925Pj + 15-69GP; - 0'8082Pj + 0'0256P; - 0'0005Pr + . . . 

 and 



1-9610P5 - Q'06823 P; + 0'001G4P; - 0'000025P^ + . . . 



respectively, giving for the ratio of the tide-height to the equilibrium tide-height at 



the equator the values 



- 1-5016, 234-87, + 2-1389. 



When the depth of the ocean is greater than 58,080 feet the tides are therefore 

 direct at the equator. They gradually increase in magnitude as the depth decreases, 

 and become infinite and change sign for some critical value of the depth rather in 

 excess of 29,040 feet after which, for further decrease of the depth, they remain inverted 

 until a second critical value is reached which is somewhat greater than 7260 feet, 

 when a second change of sign occurs. The very large coefficients which appear when 

 hg/ia)"a~ = YO indicate that for this depth there is a period of free oscillation of semi- 

 diurnal type whose period differs but slightly from half-a-day. On reference to the 

 tables of 9 it will be seen that we have, in fact, evaluated this period as 12 hours 

 1 minute, while for the case %/4ar 3 = ^ we have found a period of 12 hours 

 5 minutes. We see then that though, when the period of the forced oscillation differs 

 from that of one of the types of free oscillation by as little as one minute, the forced 

 tide may be nearly 250 times as great as the corresponding equilibrium tide, a 



* The zonal harmonics from P, up to P 7 have been tabulated by GLAISHEU, 'Brit. Assoc. Reports,' 

 1879, but I do not know of the existence of any Tables of the Tesseral Harmonics, with the exception 

 of a few given by THOMSOX and TAIT, 'Nat. Phil.,' vol. 2, 784. 



