12 MR. AIRY ON THE LAWS OF THE TIDES 



a. COS n— 2 J, a. cos n—lJ, a. cos n&, a. cos n-\-\J, a.cos w+2.^, &c., where n increases 

 by unity for each successive high water. If we take tlie 4th, the 8th, the 12th, &c. 

 differences of these numbers, we shall have for the differences standing opposite to 

 a.cosn&j 



a.cosw^X 16 sin^g"' 

 «.cos w^X256 sin^y 

 a .cos w^ X 4096 sini2_. 



Now if the inequality occupies many tides in going through its changes, that is, if 

 is small, the powers of sin ^ will be very small, and these differences will therefore 

 become smaller and smaller till they are nearly insensible. There is one value of ^, 



6 



however, for which they do not become smaller, namely, that which makes sin^ 



nearly =1, or nearly =180°, or in which the successive numbers a.cosn—\.0, 



a.cosnd, a.cosn-{-l.0, &c. have nearly equal magnitudes with a change of sign at 



every step. It is evident that this is the case of diurnal tides. Consequently, on 



taking the successive differences in this manner, the diurnal tide will ultimately be 



the only inequality sensible. 



If then we stop at the fourth differences, we may say that the diurnal tide 



fourth difference .„ , ^ ,, - ij.i. i-rn ^, j. ^ ^. ^ eighth difference 

 _ : if we stop at the eighth difference, the diurnal tide =-^ 2 ; 



16 sin*- 256 sin^j- 



and so on, the expressions becoming more accurate as we advance further in the 

 order of differences. Remarking, however, that the diurnal tide goes through all its 

 changes in not fewer than 57 high waters, and that & therefore differs from 180° by 



9 1 



little more than 6 , or that sin2 = cos 3° nearly =1 — — nearly, we may consider the 



powers of sin^ as equal to unity ; and thus we have 



Diurnal tide =jgX 4th difference, 



or =2^ X 8th difference, 

 &c. 



The first of these formulae was used throughout, both for heights and for times, and 

 at both high and low waters. 



Let us now consider the relation between the diurnal tide in height and that in 

 time. Let 6 be an angle increasing uniformly with the time, and increasing by 360° 

 in a tidal day, its origin being the time of high water in the semidiurnal tide. Let a 



