238 Sir William Thomson on an Alleged Error 



latitude in each hemisphere, and direct tides in the north and 

 south polar areas beyond. This is the state of things for any 

 value of e greater than the second critical value just considered 

 and less than a third. When e is increased through this third 

 critical value, a third pair of nodal circles grows out from the 

 poles; and there are inverted tides at the equator, direct tides 

 in the zone between the nodal circles of the first and second 

 pair, inverted tides in the zones between the second and third 

 nodal circles of each hemisphere, and still, as in every case, direct 

 tides in the areas round the poles ; a fourth critical value of e 

 introduces a fourth pair of nodal circles, and so on. 



17. The critical values of e which we have just been consider- 

 ing are of course those corresponding to depths for which free 

 vibrations of the several types described are symperiodic with the 

 disturbing force ; and the free oscillations without disturbing 

 force are in these cases expressed by the formula (3) of § 5, 

 with K 2 = — that is to say, by 



a = K 4 cc 4 -f K 6 # 6 + K 8 # 8 + &c, 



where K 4 , K 6 , &c. are to be found by giving an arbitrary value 

 to any one of them, and determining the ratios R 1? R 2 , &c. by 

 successive applications of Laplace's formula, 



with diminishing values of k, commencing with a value corre- 

 sponding to the highest ratio to be used in calculating coefficients 

 in the series. If we thus find R 2 =§, the next application of 

 the formula gives R 1 = oo , which is the test that the value of e 

 used in the calculation corresponds to a depth for which the 

 period of one of the free oscillations is exactly half the earth's 

 period of rotation. 



18. The calculation of the ratios R )} R 2 , R 3 is an exceedingly 

 curious and interesting subject of pure mathematics or arith- 

 metic. First, remark the rapid extinction of the error resulting 

 from taking or any other than its true value for R fc+ i in 

 the first application of the formula (8). Supposing k to be so 



large that j—r — ^r is a small fraction, we know that this is 



somewhat approximately the value of R p and that - , , 



is still more approximately the value of Rfc +1 . Hence we see at 

 once how small the error is if we take instead of R* +1 . If we 

 take +oo for R^+i, the formula gives for R*, and then rapid 



