298 REPORT— 1876. 



" (3) The lunar fortnightly or solar semiannual tide is a variation on the 

 average height of water for the twentj-four lunar or the twentj^-four solar 

 hours, according to which there is on the whole higher water all round the 

 equator and lower water at the poles, when the declination of the disturbing 

 body is zero, than when it has any other value, whether north or south ; and 

 maximum height of water at the poles and lowest at the equator, when the 

 declination has a maximum, whether north or south. Gauss's way of stating 

 the circumstances on which 'secular' variations in the elements of the solar 

 sj^stem depend is convenient for explaining this component of the tides. Let 

 the two parallel circles of the north and south declination of the moon and 

 anti-moon at any time be drawn on a geocentric spherical surface of radius 

 equal to the moon's distance, and let the moon's mass be divided into two 

 halves and distributed over them. As these circles of matter gradually vary 

 each fortnight from the equator to maximum declination and back, the tide 

 produced will be solely and exactly the ' fortnightly tide.' 



" 810. In the equilibrium theory as ordinarily stated, there is at any place 

 high water of the semidiurnal tide precisely when the disturbing body, or 

 its opposite, crosses the meridian of the place ; and its amount is the same 

 for all places in the same latitude ; being as the square of the cosine of the 

 latitude, and therefore, for instance, zero at each pole. In the corrected 

 equilibrium theory, high water of the semidiurnal tides may be either before 

 or after the disturbing body crosses the meridian, and its amount is very 

 different at different places in the same latitude, and is certainly not zero at 

 the poles. In the ordinarily stated equilibrium theory, there is, lirecisely at 

 the time of transit, high water or low water of diurnal tides in the northern 

 hemisphere according as the declination of the body is north or south ; 

 and the amount of the rise and fall is in simple proportion to the sine of 

 twice the latitude, and therefore vanishes both at the equator and at the 

 poles. In the corrected equilibrium theory, the time of high water may be 

 considerably either before or after the time of transit ; and its amount is very 

 different for different places in the same latitude, and certainly not zero at 

 either equator or poles. In the ordinary statement there is no lunar fort- 

 nightly or semiannual tide in the latitude 35° 16' (being sin-^ — -), and its 



\/3 



amount in other latitudes is in proportion to the deviations of the squares of their 

 sines from the value g. In the corrected equilibrium theory each of these tides 

 is still the same in the same latitude, and vanishes in a certain latitude, and in 

 any other latitudes is in simple proportion to the deviation of the squares of their 

 sines from the square of the sine of that latitude. But the latitude where 



there is no tide of this class is not sin-i— — , but sin-i( v/— il — ), where <it is 



V 3 3 



the mean value of 3 sin'Z — 1 for the whole covered portion of the earth's 

 surface, a quantity easily estimated by a not very laborious quadrature, from 

 sufficiently complete geographical data of the coast lines for the whole earth. 

 " As the fortnightly and semiannual tides most probably foUow in reality 

 very nearly the equilibrium law, it becomes a matter of great importance to 

 evaluate this quantity ; but we regret that hitherto we have not been able to 

 undertake the work. Conversely, it is possible that careful determination of 

 the fortnightly tides at various places, by proper reduction of tidal observa- 

 tions, may contribute to geographical knowledge as to the amount of water 

 surface in the hitherto unexplored districts of the arctic and antarctic 

 regions. 



