1878.] Observations in the Gulf of Cutch. 45 



The portion of the height of the sea-level above or below its mean 

 height (with reference to some fixed datum line), which is due to the com- 

 bined influences of the several tides produced by any one of the fictitious 

 stars, is given by the following well-known expression of the law of periodi- 

 city :— 



h = B^ cos{nt — €i) + H^ eos(2nt — Cj) + B^ cos(Snt — €3) + ... 

 in which h is the height above mean sea at any moment, t is the time ex- 

 pressed in mean solar hours, commencing at 0*^ , astronomical reckoning, 

 and n is the angular velocity of the star in degrees of arc per mean solar 

 hour, so that 360° -r- n denotes the period of the star in hours of mean 

 time. i2j is the amplitude, and e^ the epoch of the full-period tide; B^ 

 and €25-^3 ^^^ ^3' ^^'7 ^^^ ^^® amplitudes and epochs of the sub-tides, 

 whose periods are one-half, one-third, &c., that of the primary period. The 

 amplitude is the semi-diameter of the circle whose circumference indicates 

 the path of a tide. The epoch is the arc which, when divided by the angu- 

 lar velocity of the tide, gives the hour-angle when the height of the tide is 

 a maximum ; this occurs, on the day of starting, when nt = c^ for a prima- 

 ry tide, when 2nt = c^ (and again 12 quasi-honrs afterwards) for a tide 

 whose period is half that of the primary, and so on. 



Thus, if we now put h for the height of the sea-level at any moment, 

 and A for the value of the height of the mean sea-level which results from 

 the combined influence of the whole of the fictitious stars, we have — 



7i = A + 'Z \ Bi cos{nt — e,) + B^ cos{2nt — e^) + . . . ( 



where the symbol S stands for the summation of the whole of the terms 

 within the brackets, which relate to all the fictitious stars. 



There are two principal stars, respectively called S and M for brevity, 

 the first of which represents the mean sun, or that point in the plane of the 

 earth's equator whose hour-angle is equal to mean solar time ; the second 

 represents the mean moon, a point moving in the plane of the equator with 

 an angular velocity equal to the mean angular velocity of the moon. The 

 other fictitious stars respectively furnish the corrections to S and M for 

 declination and parallax, to M for lunar evection and variation, and to S and 

 M for the compound actions which produce what are called Helmholtz 

 Tides, &c. The 24th part of the period of star S being an hour, that of 

 any other of the fictitious stars may be conveniently spoken of, and is here 

 called a quasi-hour. 



To find the argument (the angular velocity n of the preceding formu- 

 lae) for each fictitious star, various combinations have to be made of the 

 following fundamental angular velocities, viz. : — 



7, the earth's rotation = 15°-041068G per mean solar hour. 



<r, the moon's revolution round the earth . . = 0*54901G5 ,, „ 



ju, the earth's revolution round the sun ... . = 0-0410G86 „ „ 



u), the proynission of the moon's perigee ., = 0004G418 „ „ 



