178-179] TIDE IN EQUATORIAL CANAL. 289 



where rj is the elevation given by the ' equilibrium-theory,' viz. 



(7), 

 and o- = 2ri, <T O = 2c/a. 



For such moderate depths as 10000 feet and under, n' 2 a 2 is large 

 compared with gh- 3 the amplitude of the horizontal motion, as 

 given by (4), is then //4rc/ 2 , or gf^n'^a . H, nearly, being approxi- 

 mately independent of the depth. In the case of the lunar 

 tide this is equal to about 140 feet. The maximum elevation is 

 obtained from this by multiplying by 2 h/a ; this gives, for a depth 

 of 10000 feet, a height of only 133 of a foot. 



For greater depths the tides would be higher, but still inverted, 

 until we reach the critical depth n'^cfijg, which is about 13 miles. 

 For depths beyond this limit, the tides become direct, and 

 approximate more and more to the value given by the equi- 

 librium theory*. 



179. The case of a circular canal parallel to the equator can 

 be worked out in a similar manner. If the moon's orbit be still 

 supposed to lie in the plane of the equator, we find by 

 spherical trigonometry 



cos ^ = sin 6 cos (n't H -- ? z + e } ............ . . .(1), 



V a sm 6 ) 



where 6 is the co-latitude, and x denotes the distance of any point 

 P of the canal from the zero meridian. This leads to 



X = - d ^ - = -/sin sin 2 (n't + ?-* + e] . . .(2), 

 dx \ asmO J 



and thence to 



Hence if n'a > c the tide will be direct or inverted according as 

 6 ^ sin" 1 c/n'a. If the depth be so great that c > n'a, the tides 

 will be direct for all values of 6. 



If the moon be not in the plane of the equator, but have a co-declination 

 A, the formula (1) is replaced by 



cos ^ = cos 6 cos A + sin 6 sin A cos a ........................ (i), 



* Cf. Young, I. c. ante p. 270. 

 L. 19 



