Decembek 7, 18.S3.] 



SCIENCE. 



745 



liLACIAL SCRATCUEr^ 



but llie motion clianged to north-west about 

 the niukilc line of the group. The great variety 

 of rocks in north and south strips gives abun- 

 dant opportunity for determining this motion by 

 tlic direction of dispersion of tiie bowlders from 

 their parent ledges. Iso Scottish bowlders are 

 found here, nor do marine remains occur in the 

 drift. Raised beaches do not appear on any 

 of the islands. It is concluded tiiat .Scandina- 

 vian ice covered Shetland, while Scottish ice 

 advanced over the Orknej-s ; the original mo- 

 tion of both glacial sheets being changed where 

 tliey coalesced, in the shallow Js'orth Sea, and 

 turned to the line of least resistance. — north- 

 west to the open ocean. There tho_v must 

 have ended in a great ice-dilf like that dis- 

 covered by Ross in the Antarctic Ocean. It 

 may be well to refer here to Ilelland's stud}- 

 of the Faroes a few years ago, when he showed 

 that thej" bear no marks of continental glacia- 

 tion, the few scratches he found there dei)end- 

 ing on local form for their guidance. 



Our space forliids mention of the many other 

 interesting topics that Mr. Tudor's book dis- 

 cusses, although few \olumcs contain so many 

 pages of entertainment to the general reader ; 

 but attention should be called to the well- 

 considered character of the work, only sel- 

 dom marred by a remnant of newspaper style. 

 In its table of contents, illustrations, glossary, 

 bibliograi)hy, and index, the volume is all that 

 can be desired. 



WEEKLY SUMMARY OF THE PROGRESS OF SCIENCE. 



MATHEMATICS. 

 Partial differential equations. — M. Darboux 

 considers an arbitrary partial diffoicntial equation, 

 defining a function, 2, of any number of variables. 

 Replacing z by 2 + ez', developing according to pow- 

 ers of e, and equating to zero the coefficient of c, a 

 new equation is formed, which the author calls tlie 

 auxiliary equation. The auxiliary equation defines 

 solutions differing infinitely little from a given solu- 

 tion; and so it luis a signification which does not 

 depend on the choice of variables, and wliicli will 

 remain unchanged by any arbitrary change of the 

 variables. The equation, being linear, is easy to deal 

 with, and conducts to many important results which 

 are intimately connected with the given equation. 

 The autlior considers especially two geometrical prob- 

 lems. First: having given a surface, 2, attempt to 

 find all tlie infinitely near surfaces which will form 

 with S one family of a trijily orthogonal system. 

 This problem, which has already been studied by 

 Prof. Cayley, is equivalent to either of the following 

 problems: 1°, To find all surfaces admitting of the 



same splierical representation as S; or, 2°, To find 

 all the systems of circles normal to the family of sur- 

 faces of which 2 is one. It follows at once, that, if 

 the problem of the spherical lepresentation of 2 is 

 solved, the solution can be at once arrived at for the 

 inverse surfaces to S, or the surfaces arrived at by 

 the transformation by reciprocal radii. 



The second problem considered by M. Darboux is 

 one famed for its extreme dlflSculty; viz., to find the 

 surfaces applicable to a given surf.ace. Denote by <!x, 

 i!)/, iz, the increments taken by x, y, z. in passing from 

 a point of the given surface, 2, to the corresponding 

 point on an infinitely near surface: then, expressing 

 the necessary condition to the solution of the prol)- 

 lem, — viz., that tlie small arc shall not change its 

 length, — we have — 



dx d . Sx, -I- dy d . Sy + dz d . 6z —O. 

 Replacing ix, etc., by proportional quantities, — say, 

 ^1, 2/ 11 2i, — tins is dzdx, + dy dy ^ + dzdz, = 0; i.e., 

 the corresponding elements on the surfaces 2 and 2, 

 are orthogonal. M. Darboux's problem is thus con- 

 ducted back to a problem solved Ijy M. Mi utard. The 



