426 



SCIENCE. 



[Vol. I., No. 15. 



desert, are not considered. Most of these are 

 known as recent shells also ; but we think it 

 would have been a useful addition to the collec- 

 tion if such extinct forms as Trj'ouia had been 

 included. 



To the catalogue in tabular form and the ex- 

 planation of the plates are prefixed a statement 

 of the object and method of the work, a re- 

 sume of the subject by zoological families, and 

 some general considerations. In these last the 

 author, we think, is the first to enunciate certain 

 propositions, which, though simple, constitute 

 an important advance over previous statements 

 of the general topic. In brief, he points out 

 the high probabilitj^ that lacustrine, at first 

 brackish-water forms, were derived from ma- 

 rine species by imprisonment due to rising 

 seashores, forming, first estuaries, then lakes ; 

 afterward differentiated ao as to become inured, 

 to water without salt, or, in other cases, exter- 

 minated by water too fresh, or in lakes without 

 an outlet, hy concentration of saline matters. 

 This view is not whollj' novel ; but the author 

 goes on to supplement it by pointing out how, 

 from the gradual conversion of lakes into rivers, 

 and the persistence of the latter through epochs 

 of geological change, the remarkable persisten- 

 cj- of fluviatile types is accounted for, and prob- 

 lems of present geographical distribution maj- 

 be solved. 



Too much space would be required for an 

 analysis of the work in detail : a few points 

 have been noted for mention. It seems a 

 little startling to have 03'sters, Anomiae, and 

 mussels presented as non-marine, until we learn 

 that they were of the peculiar brackish-water 

 beds in the Laramie group, and were doubtless 

 accustomed to almost lacustrine conditions. 

 The oldest forms treated of are Naiadites and 

 certain supposed Anodontae from the Devonian 

 (the latter much suggesting in appearance 

 Lithodomus and its allies) ; but if these were 

 not, as is supposed, true fresh-water folk, then 

 the earliest of the latter date I'rom the trias. 

 Six families of Conchifera, in all, aud sixteen 

 of Gastropoda, are represented in the catalogue. 

 It might be suggested that an analogue of Unio 

 belliplicatus may, perhaps, be found living in 

 Nicaragua, and that C'erithidea lives rather 

 abundantly' on the Californian coast. 



The proof-reading of this volume is not up 

 to the usual standard of the Go-\'ernment print- 

 ing-office, and the index is disappointingly 

 meagre. The arrangement of the numbers to 

 figures on the plates is confused and puzzling : 

 it can hardlj' fail to cause a serious loss of time 

 to those who consult them. On the other hand, 

 the paper and press-work are above the aver- 

 age, and the execution of the figures unusuallj' 

 good. 



WEEKLY SUMMARY OF THE PROaRESS OF SCIENCE. 



GEODESY. 

 Geodetic night-signals. — Mr. C. O. Boutelle, of 

 the U. S. coast-survey, finds tliat the magnesium light 

 as used by the sui'vey may be used for distances as 

 great as forty-five to seventy miles, and that the ordi- 

 nary student-lamp with a parabolic reflector may be 

 seen as far as forty miles. A report on night-sig- 

 nals was published by the coast-survey last year. 

 The advantages stated in the report, as derived from 

 greater steadiness of the atmosphere, and comparative 

 freedom from lateral refraction upon long lines of 

 sight during night observations, have been signally 

 verified during the seasons of 1881 aud 1882. — (Rep. 

 U.S. coast cjeod. sure., 1880.) [859 



MATHEMATICS. 



A definite integral. — In a brief note, M. Kor- 

 kines gives a simple proof of a theorem due to M. 

 Tchebychef. The theorem relates to the integral 



J(j)(x] i'lx) dx, where (f and V must satisfy one of the 

 

 two conditions: 1°, they simultaneously increase or 

 simultaneously decrease for all values of x lying 

 between zero and unity; 2°, or one of them must 

 increase and the other decrease for the same values 

 of X. In the first case, M. Tchdbychef's theorem is 



j\{x) t{x] dx > jr|»(.T) dxjn^) dx; 



in the second case, 



ffix) i>(x) dx < j^(/.(.x) dx Jf{x) dx. 



M. Korkine makes these theorems the immediate 

 consequence of a simple identity. — (Comptea rendus, 

 Jan. 29. ) T. c. [860 



Linear differential equations. — In a ijrevious 

 communication to the academy, M. Goursat has 

 solved, for a special class of equations, the problem 

 to find the entire number of substitutions to which 

 a system of fundamental integrals of a given equa- 

 tion may be subjected, corresponding to all the differ- 

 ent closed paths which the variable may describe. 

 The general integral in that case was shown to be 

 expressed by liypergeometric series of higlier orders. 

 In the present paper, M, Goursat develops more fully 

 his method, and applies it to the equation of the third 

 order, remarking that the metliod followed is iden- 

 tically the same for equations of any order. — 

 {Comptes rendus, Jan. 29.) T. c. [861 



Functions of two independent variables. — 

 M. Picard has given a series of notes upon this sub- 

 ject, determining the functions of two independent 

 variables, u and v, which remain invariable when we 

 effect upon u and v any of the infinite number of 

 substitutions of a linear discontinuous group. In 

 the present paper Jl. Picard consider,*, in a general 



