TRANSACTIONS OF SECTION A. 541' 



points, the remaining four points of intersection of the surfaces at/z+f.iw = 0, 

 bzx + gyiu = 0, cxy + hzto = 0, and the remaining four points of intersection of the 

 surfaces ayz —fxiv = 0, bz.T—ffyw = 0, cxy — hzw = 0. 



The above is the analytical theory of one of the two forms of quartic sui'face 

 •with twelve nodes recently established by Dr. K. Rohn in a paper in the 'Abhand- 

 lungen der K. Sachsische Gesellschaft zu Leipzig.' 



7. 



On the Jacohian Ellifsoid of Equilibrium of a rotating Mass of Fluid. 

 Bij Professor G. H. Darwin, F.B.S. 



8. On the Dynamical Theory of the Tides of Long Period. By Professor 



G. H. Darwin, F.B.S. 



9. Note on Sir William Thomson^ s Correction of the Ordinary Equilibriti/m 

 Theory of the Tides. By Professor J. C. Adams, LL.D., F.B.S. 



In Art. 806 of Thomson and Tait's ' Treatise on Natural Philosophy,' it is 

 pointed out that if the earth's surface is supposed to be only partially covered by 

 the ocean, the rise and fall of the water at any place, according to the equilibrium 

 theory, would be falsely estimated if, as is usually done, it were taken to be the same 

 as the rise and fall of the spheroidal surface that would bound the water were there 

 no dry land. 



In the articles which immediately foUow the above, it is shown that in order to 

 satisfy the condition that the volume of the water remains unchanged, the expres- 

 sion for the radius vector of the spheroid bounding the water must contain, in 

 addition to the terms which would be sufficient if there were no land, a quantity 

 (a) which depends on the positions of the sun and moon at the time considered, 

 and which is the same for aU points of the sea at the same time. 



This quantity (a) contains five constant coefficients which depend merely on the 

 configuration of land and water. The values of these coefficients in the case of the 

 actual oceans of our globe have been carefully determined very recently by Mr. 

 H. H. Turner, of Trinity College, in a joint paper by Professor G. H. Darwin and 

 himself, which is published in vol. xl. of the ' Proceedings of the Royal Society.' 



It should be remarked that every inland sea or detached sheet of water on the 

 globe has in the same way a set of five constants, peculiar to itself, which enter into 

 the expression ot the height of the tide at any time in that sheet of water. 



By taking such constants into account the formulae which apply to the oceanic 

 tides are rendered equally applicable to the tides of such a sea as the Caspian, which 

 are thus theoretically shown to be very small, as they are knovpn to be practically. 



In the work above cited reference is made to a passage in a memoir by Sir 

 William Thomson on the Rigidity of the Earth, published in the * Philosophical 

 Transactions ' for 1862, as being the only one known to the writers in which any 

 consciousness is shown that such a correction of the ordinary equilibrium theory as 

 that above mentioned is required. 



However just this remark may be in reference to modem writers on the equili- 

 brium theory, it is only fair to Bernoulli, the originator of the equilibrium theory, to 

 point out that in his prize essay on the tides he distinctly recognises the fact that 

 when the sea is supposed to have only a limited extent the rise and fall of its 

 surface cannot be the same as if the earth were entirely covered by it. In parti- 

 cular, he shows that the tides are so much the smaller as the sea has less extent in 

 longitude, and thus explains why they are altogether insensible in the Caspian and 

 in the Black Sea and very small in the Mediterranean, of which the communica- 

 tion with the ocean is almost entirely cut off at the Strait of Gibraltar (see 

 Bernoulli, 'Traits sur le Flux et Reflux de la Mer,' Chapitre XL section ii. 



It may be as well to mention that this treatise of Bernoulli, as well as the dis- 

 sertations of Maclaurin and Euler on the same subject, is published in the 3rd 

 volume of the Jesuits' edition of Newton's 'Principia,' and also appeai-s in the 

 Glasgow reprint of that edition. 



