178-179] TIDE IN EQUATORIAL CANAL. 289 



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



2 \ a J 



and a = 2ri t &amp;lt;T O = 2c/a. 



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

 compared with gh; the amplitude of the horizontal motion, as 

 given by (4), is then //4/i /2 , or g/4tn 2 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 2 a 2 /g, 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 



/ x \ 



cos S- = sin 8 cos [rit + r. + e (1), 



V a sm 6 ) 



where 8 is the co-latitude, and x denotes the distance of any point 

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



X = - j- = -/sin 8 sin 2 (n t + X a + e) . . .(2), 

 dx \ asin.0 J 



and thence to 



cos 2 



(n t + ^7,+ e) ...... (3). 



V a sin 6 ) 



2 c 2 - n 2 a 2 sin 2 8 



Hence if ria &amp;gt; c the tide will be direct or inverted according as 

 8 ^ sin&quot; 1 c/n a. If the depth be so great that c &amp;gt; ria, the tides 

 will be direct for all values of 8. 



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 



