184 MR. S. S. HOUGH ON THE APPLICATION Of HARMONIC 



agreeing exactly with twelve hours, when the polar depth is 36,970 feet, the 

 equatorial depth 26,290 feet, and the mean depth 29,517 feet. 



18. Diurnal Tides. 



It has been shown in 14 that the diurnal tidal constituents whose periods are 

 equal to a sidereal day will involve no rise and fall at the free surface when the 

 depth of the ocean is uniform. This theorem will be rigorously applicable to the 

 luni-solar diurnal constituent usually designated by the initial K l9 while it may also 

 be supposed to apply with a fair degree of accuracy to each of the solar diurnal 

 constituents since the motion of the sun in his orbit is sufficiently slow. There will 

 however be an important lunar diurnal constituent for which the speed is 0'92700w, 

 in dealing with which we propose to take into account the difference between the 

 period and a sidereal day. The method of computation is exactly similar to that 

 used for the lunar semi-diurnal tides, and thus we find when hg/lurtC 1 = -f , 



= i[ 0-07G38PI + 0-03543P1 - 0'00845Pi + 0'001207PJ 



- 0-000114P! + 0'000008PJ 3 - . ..]; 



when /i<7/4ora 3 = - 2 J -, 



= @.![- O'lG'JlP.l + 0-04738PJ - 0'0062SPi + 0'000480PJ 



0-000024PJO + O'OOOOOl Pj, . . .] ; 

 when hy/ihr'a" = -^, 



= Cf.i[- 0-4 145 PI + 0-06576PJ 0-00464PJ + 0'000184PJ - O'OOOOOSPJo + ...]; 



and when hy/4ara? = 5, 



= <![- 1-4428 PI -1- 0-1231PJ - 0'00449PJ + 00009 PJ - O'OOOOOlPJo +...]. 



It appears, then, that these tides will increase with "the depth, and that they will 

 be in the main inverted. For small depths the rise and fall will be small, but with 

 a depth as great as 58,080 feet the tide will be in excess of the equilibrium-tide. 

 The type will tend to approximate more and more closely to that represented by 

 a second order harmonic alone as the depth increases. 



So long as the depth is uniform the tidal constituents whose periods are rigorously 

 equal to a sidereal day will never tend to become infinite, and consequently no 

 period of free oscillation of the type of the diurnal tides can coincide exactly with 

 one day. As w diminishes however the shortest period of the second class, which for 

 the depths under consideration is longer than the period of the lunar diurnal 



