236 Sir William Thomson on an Alleged Error 



{e =10), « = H{l-0000.a? 2 + 20-1862.a? 4 + 10-1164..r 6 



- 13-1047 . x 8 - 15-4488 . x ]0 - 7*4581 . x n 

 -2-1975 . a 14 -0-4501 . x 16 -00687 . .r 18 

 -0-0082. # 20 -0-0008 .^ 22 -0-0001 .# 24 }; 

 (c=2-5), a = H {1-0000. x* + 6-1960. a> 4 + 3-2474. # 6 



+ 0-7238. # 8 + 0-0919. x l0 + 00076. x™ 

 + 0-0004. x 14 }; 

 (e= 1-25), « = H{ 1-0000. a; 2 + 0-7504. x 4 + 0-1566. .r 6 

 + 0-01574. a? 8 + 0-0009. a? 10 }. 



By putting # = in each case we find a=0, showing that there 

 is no rise and fall at the poles. Putting a? = l, we find in the 

 three cases, 



a— — 7*434 . H . . . (depth ^— of radius), 

 a= 11-267. H...( „ ^L_ M }j 



0= i«54.H...( „ 3^-. „ ). 



The negative sign in the first case shows that the tide is " in- 

 verted " at the equator; or there is low water when the disturb- 

 ing body is on the meridian, and high water when it is rising 

 or setting. For small values of x (that is to say, for polar 

 regions) the sign is positive, and therefore the tides are direct 

 for this, as clearly for every other depth (because in every case 

 the first term is + H# 2 ). In the particular case in question 

 (depth ] ), as we see from the formula given above, the value 

 of a increases from zero to a positive maximum, and then de- 

 creases to the negative value stated above as x is increased from 

 to 1 ; and the intermediate value of x which makes it is 

 roughly *95, or the cosine of 18°. Hence Laplace concludes 

 the tides are inverted in the whole zone between the parallels of 

 18° north and south latitude, while throughout the regions 

 north and south of these latitudes the tides are direct. The 

 formulae given above for the second and third of the depths 

 chosen by Laplace shows that in these cases the tides are every- 

 where direct and increase continuously from poles to equator. 



The results of the summation for the equatorial tide in the 

 three cases given above are very interesting as showing how 

 much greater it is in each case than H (the equilibrium height) . 

 Upon this Laplace remarks that for still greater depths the value 



