Decembei: l'I, IS*j.l 



SCIENCE. 



199 



that the theory needs correction to take account 

 of the relative amount of land and water, as 

 ■well as the contour of the continents. These 

 have a controlling; influence upon the tides, and 

 this discovery is Thomson's great improvement 

 and correction of the equilibrium theory. 



The diurnal tide has been usually explained, 

 in accordance with the equilibrium theory, as a 

 wave existing under nearl}- static conditions, 

 and following the moon and sun around the 

 earth, but interfered with bj- friction, and 

 changed in direction b}' the contour of the 

 land. Though this was the view of Newton, 

 Young, and others, and is incorporated in our 

 ordinary text-books, it is quite inadequate ; 

 and the kinetic theor\- of Laplace must be put 

 in its place, which treats the water as a moving 

 fluid body, subject to the disturbing influence 

 not only of the sun and moon, but of itself 

 also. 



The kinetic theory of the tides was to have 

 been developed at length in vol. ii. ; and that 

 intended development is more than once re- 

 ferred to by the authors, — as, for instance, on 

 p. 382, where an incidental comparison is made 

 of the results of the two theories. 



This part of the theory has been treated by 

 Ferrel in his ' Tidal researches,' published as 

 one of the appendices to the U. S. coast-survej' 

 report for 1874, in which work he has put in 



practical shape all the theoretical work hereto- 

 fore accomplished, and also deduced therefrom 

 important consequences. Until the publication 

 of this work, it was not possible to apply the 

 correct theory to the discussion aiul prediction 

 of tides by reason of the unmanageable formu- 

 lae emplo3-ed by Laplace ; and the discussions 

 were, perforce, made by some modification of 

 the equilibrium theory. Indeed, Laplace him- 

 self resorted to that method in his famous dis- 

 cussion of the tidal observations in the harbor 

 of Brest. But, thanks to Ferrel's labors, this 

 most intricate branch of computation has been 

 systematized, and applied to an extensive series 

 of tidal observations in Boston harbor. 



The concluding pages, from 42"2 to 4G0, treat 

 the question of the rigidity and solidity of the 

 earth as a whole, especially as related to the 

 tides. The final sentence (p. 4G0) is, "On 

 the whole, we ma}- fairly conclude, that, whilst 

 there is some evidence of a tidal yielding of 

 the earth's mass, that yielding is certainly 

 small, and that the etteetive rigidit}' is at least 

 as great as that of steel." 



Four important papers on subjects related 

 to those just mentioned are added to the work 

 as appendices. The titles of these papers are, 

 ' Cooling of the earth,' ' Age of the sun's 

 heat,' ' Size of atoms,' 'Tidal friction.' The 

 last three of these were not in the first edition. 



WEEKLY SUMMARY OF THE PROGRESS OF SCIENCE. 



MATHEMATICS. 

 Fuchsiau functions. — A previous paper by M. 

 Poincar^ on this subject has already been noticed in 

 these pages (i. 535). In the present most important 

 memoir, M. Poincar^ assumes the results arrived at 

 in the former memoir, and proceeds to more fully 

 develop them and the consequences flowing from 

 them. In the previous paper the author showed that 

 it was possible to form discontinuous groups by sub- 

 stitutions of the form 





by choosing the coefficients a,-, ^i, y(, <J,, in such a way 

 that the different substitutions of the group sliould 

 not alter tliroughout the interior of a certain circle 

 called the fundamental circle. In the present paper 

 the autlior assumes tliat the fundamental circle has 

 its centre at the origin, and its radius unity; so that 

 its equation can be written as mod. 2 = 1. 



He then considers one of tliese discontinuous 

 groups, which he calls Fuchsian groups, and which 

 he denotes by G. To this group corresponds a de- 

 composition of tlic fundamental circle into an infinite 

 number of normal polygons, E, all congruent among 



themselves. The author tlien demonstrates that there 

 always exists a system of uniform functions of z, 

 which remain unaltered by the different substitutions 

 of the group G, and which he calls Fuchsian func- 

 tions. M. Poincar^'s memoir is too long to be re- 

 viewed here as it deserves. It is certainly a most 

 important addition to the modern theory of functions, 

 and is rendered particularly valuable by the liistorical 

 note at the end, in which the author gives a brief 

 account of the labors of Herraite, Fuclis, Klein, 

 Schwarz, and others in this field. The two memoirs, 

 with very little amplification, would constitute a 

 really valuable treatise on this subject, — a subject 

 of great importance, and on which there exists abso- 

 lutely no text-book or treatise of any kind. — (Acta 

 math., i.) t. c. [506 



ENGINEERING. 

 Steam-Tvhistles. — Lloyd and Symcs give a state- 

 ment of experiincmts with a locomotive whistle hav- 

 ing a bell 4 Is inches di.ameler, oj inches long inside, 

 and over an annular steam opening V,; of an inch 

 wide. The bell w as of cast brass of medium charac- 

 ter; and tlie lip was chamfered to a thin edge, and set 

 exactly over the steam-opening. Sixty pounds press- 



