Laplace's Theory of the Tides. 185 



The great improbability of some of Laplace's results obtained 

 with his value of K 4 may be seen by comparing his tide with 

 that of canals extending east and west around the globe. With 

 a depth of sea of about 5*5 miles, Laplace gets, with the moon 

 and sun in conjunction on the equator, a direct tide with a 

 range at the equator of 36 feet. If we now suppose the whole 

 globe to be divided into canals extending east and west around 

 the earth, say, one or two degrees in width, so that the oscil- 

 lations in the canals would be entirely independent of one 

 another, we know that with a depth of 5 "5 miles all the canals 

 between the latitudes of about 60° north and south would have 

 inverted instead of direct tides, with a range at the equator 

 less than 6 feet instead of 36 feet, and those only between 

 those parallels and the poles would have direct tides. Hence 

 the tides in the canals are inverted over the whole globe ex- 

 cept comparatively small areas between the parallels and 

 the poles. If we now suppose the partitions between the 

 canals to be removed, the conditions, of course, would be 

 somewhat changed, and there would be north and south 

 motions of the water also besides the principal east and west 

 component ; but still it is very improbable that the effects of 

 these small polar seas with direct tides by means of any inter- 

 changing motion of the water between the different latitudes, 

 would reverse the inverted tides over the whole remaining part 

 of the globe, and cause them to be direct tides with a range 

 at the equator of 36 feet, as given in Laplace's results. 



With K 4 = I have obtained in this case (Tidal Researches, 

 § 160) a tide with a range at the equator of only about 4 feet, 

 the tide being an inverted one, as in the case of the canals, 

 over the greater part of the globe, and having somewhat the 

 same range, instead of a tide completely reversed and with a 

 range at the equator of 36 feet, as given by Laplace's value of 

 that constant. 



Again, Laplace's values of K 4 make the tide at the equator, 

 as the depth is increased, change from an inverted to a direct 

 tide with a depth somewhere between 1*4 and 5*5 miles, say 

 3*5 miles, as may be seen from Laplace's results for different 

 depths as given by Sir William Thomson (§ 15). But in 

 the case of the canals this change from inverted to direct tides 

 occurs at the equator with a depth of 14 miles ; and with a 

 depth of 10'5 miles they are still inverted as far as the parallels 

 of 30°. 



With K 4 =0 my expression of a gives for the depth of 11 

 miles a very small tide at the equator with a range of only 1*2 

 foot, the tide being still inverted as in the canals. Hence 

 with a small increase of depth, say, with a depth of 12 miles, 



