ON THE COASTS OF IRELAND. 13 



be the diurnal tide at the first high water, h that at the first low water, c the semi- 

 range of the semidiurnal tide. And suppose a and h to be so much smaller than c 

 that their squares, &c. may be neglected. The height of the water above its mean 

 height, on the law of elevation usually assumed, will be a.cos^-}-A.sin^-|-c.cos2^. 

 This quantity will be maximum or minimum, or there will be high water or low 

 water, when — a. sin ^+6. cos ^— 2c. sin 2^=0. The first approximation to the value 

 of 6 will be obtained by considering the large term only : from this we find 



2c.sin 2^=0, from which ^=0, or =-, or =r, or =yj nearly. Substituting these values 



successively in the small terms, and supposing them liable to a correction x in the 

 large term, we have, 



For the first high water, +i — 2c.sin (0-f-2a?)=0; or, nearly, 



^--4c.r=0; whence .r=T-^ and ^=04- .r=—- 



For the first low water, —a— 2c. sin (5r+2.r)=0 ; or, nearly, 



— a+4ca?=0 ; whence ^^=4^' and ^=^+.r=|+^. 



For the second high water, — Z>— 2c.sin (2^-|-2jr)=0 ; or, nearly, 



— 6— 4ca?=0; whence ^= -J—' and &=:r-^a:=x—-r' 



4c '4c 



For the second low water, +a— 2c.sin (3^+20:') =0 ; or, nearly, 

 +a+4c.r=0; whence a: =^' and ^=Y~"r' 



It appears therefore that the diurnal equation in time at the High waters of the 

 First Division has the same sign as, and is a certain multiple of, the diurnal equation 

 in height at the Low waters of the First Division ; and that the diurnal equation in 

 time at the Low waters of the First Division has the same relation to the diurnal 

 equation in height at the High waters of the First Division ; and similarly for those 

 of the Second Division. The factor by which the diurnal tide at low water in height 



is converted into diurnal tide at high water in arc is ^ ; and, observing that -x- in arc 



corresponds to about 12'' 24" in time, the factor for converting diurnal tide at low 



.... 1. 1.1 , . , . . /. .. . 744 186 J 



water m height into diurnal tide at high water m minutes or time is 4^=^ ; and 



that by which the diurnal tide at low water in minutes of time is converted into 



diurnal tide at high water in height is ^- The same factor applies for converting 



diurnal tide at high water in minutes of time into diurnal tide at low water in height. 

 But the high and low waters of the First Division must be used together, and the high 

 and low waters of the Second Division must be used together. 



This theory cannot be expected to apply with accuracy to any place far from the 



