270 Mr. J. Nixon on the Tides in the Bay of Morecambe, 



The ray from P, initially in the direction of D, being con- 

 stantly curved downwards in its passage from the effects of 

 atmospherical refraction, will cut the vertical DC at a point 

 E lower than D by (nearly) the tangent of an angle DPE, 

 which bears a constant proportion to the true dip (or con- 

 tained arc). At E the depression of P is evidently less than 

 at D by the angle PEK = DPE, or the angle of refraction ; 

 but as P will not be seen from E in the direction EP, but in 

 a tangent to the extremity of the curved ray at P, it will ap- 

 pear elevated above P by the angle of refraction : hence the 

 apparent dip at E will be less than the true dip at D by twice 

 the angle of refraction *. 



DL being the height of D above the level of the sea, that 

 of E will be equal to DL minus DE (or the part cut off by 

 refraction). As the angle DPL is constantly half the dip at 

 D, (or half the contained arc PCD,) it follows that the angle 

 DPE will bear twice the proportion to that of DPL that the 

 refraction does to the contained arc. When the height in 

 feet of E is given, that of D may be found by increasing E by 



2 E 



-, n being the ratio of the contained arc to the refraction. 



72—2' ° 



Thus, if E be 80 feet, and the refraction y^jth of the arc, then 

 will the height of D be 80+ £p| = i feet. The true 



dip for D being 10' 37", that for E will be (10' 37" minus 

 y^th, or) 9' 33", and its apparent dip (10' 37" minus 1th, or) 

 8' 30". (The true dip for G (= 80 feet) would be 9' 30", or 

 3" less than for E.) 



Although well aware that the difference between the true 

 and apparent dip was subject to great and unaccountable 

 fluctuations, yet I flattered myself that by measuring along 

 with the depression of the horizon of the sea that of a fixed 

 object on the shore beyond, situate nearly in the same direc- 

 tion and at about an equal distance, I should then be able to 

 ascertain the deviation of the refraction at low water from its 

 amount at high water, the latter being equal to the difference 

 between the apparent dip and that calculated from the mea- 

 sured height of the eye above the sea at the breakwater f. 

 The plan, which, with one exception, appears to have been 



* G, the point of intersection of the unrefracted ray PD with a vertical 

 at a height equal to that of E, has a less extensive horizon than the latter 

 by (twice) the arc GE. D is elevated above E, yet their horizons, as that 

 of the former is not affected by refraction, are equal. 



t Any probable difference in the times of high water at the horizon 

 of the sea and at the breakwater would scarcely exceed the duration 



