22 MR. AIRY ON THE LAWS OF THE TIDES 



time of moon's transit ; and therefore, at high water, the lunar diurnal tide is always 

 in nearly the same phase, and has no variation except from the variation of its coeffi- 

 cient. The magnitude of the diurnal tide at semidiurnal high water may therefore 

 be represented by coefficient x sin |3 ; and that at semidiurnal low water by coefficient 

 Xcos^; where (3 is constant. This coefficient is proportional to the sine of the 

 moon's declination at some time previous, or (nearly) proportional to the sine of the 

 moon's right ascension for some time previous, or to the sine of the moon's hour- 

 angle from the sun altered by a constant. For the solar diurnal tide, the coefficient 

 is constant, but the phase varies every day. As the time of high water bears a nearly 

 invariable relation to the time of moon's transit, the phase of solar diurnal tide at 

 high water must depend upon the moon's hour-angle from the sun altered by a con- 

 stant, and therefore the magnitude of solar diurnal tide will be proportional to the sine 

 of the moon's hour-angle from the sun altered by a constant. Thus, putting ^ — O 

 for the excess of the moon's right ascension above the sun's, the lunar diurnal tide at 

 the time of high water will be represented by a.sin|3.sin{ c — O+A}, and the solar 

 diurnal tide at the same time will be represented by i.sin{ a — 0-|-B}; and these, 

 when added together, give a result of the same form, c.sin{ ^ — O -f-C}. And it is 

 impossible to say whether this term, as given by observation, is entirely due to one 

 or other of the two actions or to both combined ; because we have no a priori means 

 of saying what is the coefficient a or b of either of the separate terms ; or what is the 

 relation of the time of either high diurnal tide to the time of transit of the body which 

 causes it, upon which A and B will depend. Everything here said with regard to 

 semidiurnal high water applies also to semidiurnal low water; the only difference 

 being that the angles /3 and B must be increased about 90° for semidiurnal low water. 



The unknown quantities in the problem of diurnal tide are the following : — The 

 interval anterior to the time of observation for which the moon's place is to be taken 

 as governing the diurnal tide at the time of observation ; the constant coefficient by 

 which the sine of moon's declination for that anterior time is to be multiplied ; the 

 moon's hour-angle at the time of lunar diurnal high water ; and the three similar 

 quantities for the sun: in all, six unknown quantities. To determine these we have 

 only the four following results of observation (or results equivalent to these four) : 

 the time of evanescence of diurnal tide at semidiurnal high water; the maximum of 

 diurnal tide in high water, and the two similar quantities for low water. These are 

 insufficient for the determination of the six unknown quantities ; and we must try 

 how we can reduce the latter number. 



First, as the sun's declination is considered constant, the anterior interval for the 

 sun's place is unimportant. And in fact, though the sun's declination during these 

 observations (June 22 to August 25) was not invariable, yet an alteration of one day 

 in the time for which its declination was taken as ruling the diurnal tide would not 

 have been important. For the moon it would be very important. 



Secondly, it seems probable that the moon's hour-angle at the time of lunar diurnal 



