T I D 



Cor. To find the distance of high tide from the point where the water 

 is at the same height at which it would have been if there had been no 



tide, put 3 cos.2 8 1 = ; .*. cos. 6 = = cos. 54. 44'. 



V3 



4. To find the elevation and depression as before, produced by the joint 

 action of the sun and moon. 



Let m perpendicular height of high above low water, as caused by 

 the sun, n ditto arising from the moon, 6 = hour angle from the time of 

 high tide for the sun, 6' = ditto for the moon; then the elevation above' 

 the natural level is 



3 cos.2 _ 1 3 CCS. 2 & 1 



3 X H X n ; 



. 3 sin.2 2 3 shi.2 p 2 ,_ 

 aud depression is X m -] X . 



Cor. 1. If the sun and moon be in syzygy, 9 = 6' ; 

 .*. elevation = (m -f ?/) cos.* 8 ~T H ; 



' O 



and depression =: (m -f- n) sin.2 6 % (m + w). 



Hence at high water, elevation = % (jn -;- w), and at low water, de- 

 pression = y 3 (m 4- n}. 



Cor. 2. If the moon is in quadrature, elevation at S = % m % , 

 and depression at M y s m % n ; also the elevation at S above the 

 inscribed sphere m n, and the elevation at M above the same 

 n m. Hence since n is greater than m in the ratio (according to Ber- 

 nouilli) of 2^ : 1, it is plain that when the moon is in quadrature, it is 

 high water under the moon, and low water under the sun. 



Cor. 3. Supposing the sun and moon to he in any other position, and it 

 were required to find an intermediate point between them where there 



is high tide ; in this case we must take the expression *"" X m 



3 cos 2 6' 1 



H -,- X n, arid make the differential = 0, and we shall get 



m : n :: sin. 2 8 : &in. 2 0'. Hence we have only to divide an arc 

 2 (0 + &) into two part?, so that the ratio of the sines may be given; and 

 the half of each part will give 6 and 6', and thus we get the point where 

 the tide is highest. 



Cor. 4. By computing by the last Cor. the ^'s 9 and 0' for every day 

 /rom the new or full moon, we might get the time of the high tide when 

 301 



