March 23, 1883.] 



SCIENCE. 



197 



Encope) , from the campanularian hydroids ; 

 and the Trachomedusae (e.g., Liriope) and 

 Narcomedusae (e.g., Cunina), from the ' tra- 

 chj'larian ' hj'droids. The resemblances be- 

 tween the Aeraspedae and the Craspedotae, 

 and the similaritj' between the various orders 



of Craspedotae, he believes to be due to sec- 

 ondary modification, rather than to inheritance 

 by descent from a common ancestral medusa. 



He regards the Ctenophorae and the Sipho- 

 nophorae as divergent stems from the Antho- 

 medusae. 



WEEKLY SUMMARY OF THE PROaRESS OF SCIENCE. 



ASTRONOMY. 

 Astronomical applications of photography. — 

 Prof. E. C. Pickering described some photograpliic 

 work which is now being undertaken at the Harvard 

 observatory. Experiments are being made witli vari- 

 ous lenses, and on their completion it is intended 

 to take photographs of the whole visible heavens 

 north of .30° south. It is possible, also, that a map 

 will be published. Measurements of the photograph- 

 ic energy of all the brighter stars will be made, down 

 to, perhaps, the seventh magnitude. Besides this, it 

 is proposed to obtain measurements of the color 

 of the stars by using a large lens of heavy flint-glass, 

 giving as much chromatic aberration as possible. In 

 the centre a circular disk of glass will be placed, 

 slightly thinner at one edge than at the other. The 

 eiiect will be, that every star will have two images 

 placed side by side. By adjusting the sensitive-plate 

 at a certain distance from the lens the blue rays will 

 be brought to a focus ; but, in the case of the image 

 formed by the rim of the lens, the violet and ultra- 

 violet rays will be spread over so large an area as to 

 produce comparatively little effect, while in the other 

 image they will have nearly full power. By placing 

 another plate somewhat nearer the lens the violet 

 rays will be focused. A third plate will enable us to 

 focus the ultra-violet rays. By comparing, in each 

 case, the image formed by the edge of the lens with 

 that formed by the centre, a series of quantitative re- 

 sults can be obtained, which will vary according to 

 the spectrum of the star measured. By this method 

 any variations of color as well as of magnitude could 

 at once be detected. — (Amer. acad. arts nc. ; meet- 

 ing Feb. U.) [412 

 MATHEMATICS. 



Riemann's theory. — The present paper, by Prof. 

 Klein, is a continuation and generalization of the 

 methods and results in his memoir, which appeared 

 a year ago, entitled Ueber Eiemann's Theorie der 

 Algebraischen Functionen, etc. This last contained 

 an extension of the Kiemann theory of functions to 

 arbitrarily given closed surfaces. There exist over 

 these surfaces, as the author shows by physical con- 

 siderations, certan potential functions, the relations 

 between which, expressed in the language of analy- 

 sis, afford the sought properties in the theory of func- 

 tions. The physical considerations at first employed 

 in order to obtain tentative results are now aban- 

 doned, and the author develops his new theory by 

 more rigorous methods. Instead, now, of consider- 

 ing a Eiemann's surface as a closed surface, he re- 

 gards it as a bounded surface, or aggregate of bounded 

 surfaces, where the different portions of the bounding 

 curves may be regarded as being connected in pairs by 

 any assigned law. A so-bounded surface is regarded 

 as a portion of a closed surface ; and the author shows 

 how an important general principle is obtained, 

 which he calls the principle of analytical develop- 



ment, and which, in certain special cases, coincides 

 with a principle of Schwarz called the principle of 

 symmetry. The author shows how, by certain par- 

 ticularizations of the ideas, a general notion may be 

 obtained of those functions which have linear trans- 

 formations among themselves; and a theory is tlien 

 given of single-valued functions of this kind. The 

 author speaks of a Eiemann's manifold, instead of a 

 Eiemann's surface, and considers a closed two-dimen- 

 sional manifold instead of a closed surface, and, upon 

 this manifold, single-valued definite differential ex- 

 pressions, instead of simply the element of length. 

 Numerous references are given to the earlier litera- 

 ture of the subject, in which the investigations of 

 Poincar^ stand out most prominently. The present 

 memoir, taken with the previous one above referred 

 to, constitutes one of the most important additions 

 that has ever been made to Riemann's theory of func- 

 tions. — {Math, annalen, xy.\.) T. c. [413 

 Functions of t'wo variables. — M. Poincare gives 

 a generalization of a theorem of Weierstrass concern- 

 ing functions of one variable. The theorem in ques- 

 tion is, " If F{x] is a mer amorphic function over the 

 entire plane, it can be placed in the form of a quotient 

 of tioo integral functions." M. Poincar^ seeks to find 

 the analogous theorem in the case of two variables, 

 and considers a function, P (X,Y), of two imaginary 

 variables (X = x -|- iy, Y — z-\- it). Calling u the real 

 part of a function of X. and Y, it is seen that u satis- 

 fies a differential equation (Am = 0) where 



dx-'i "^ dy^ '^ dz^ dt^ ' 

 u also satisfies certain other partial differential equa- 

 tions of the second order, which need not be written 

 down. Any function satisfying the equation Am = 

 is called a potential function. The aggregate of 

 points satisfying the inequality 



(x-x„)2-f-(2/-2/o)' + (a-2o)^ + («-«o)'< »•' 

 is called a hyjierspheric region. The author con- 

 structs an infinite number of hyperspheric regions, 

 and considers a point (xy zt) as belonging to at least 

 one of these regions, and being common to not more 

 than five of them. The final theorem obtained is as 

 follows: if Y is any non-uniform function of X, — 

 which has no essential singular points at a finite dis- 

 tance, and which cannot, for the same value of X, 

 take an infinite number of values infinitely near to 

 each other, — it can be considered as the solution of 

 an equation, G- (X, Y) = 0, where G is an integral func- 

 tion. — ( Compies rendMS, Jan. 22. ) t. c. [414 



PHYSICS. 

 Mechanicsi 

 Motion of a pendulum. — M. Lipschitz, in a let- 

 ter to M. Hermite, investigates the motion of a heavy 

 body capable of turning freely about a horizontal axis. 



