18 



SCIENCE. 



[Vol. II., No. 22. 



able stars in both positions of the instrument. 

 Some forms of the ' broken telescope ' transit, espe- 

 cially those with a slender axis, require the addition 

 of terms involving other functions of z than its 

 cosine. Prof. Watson found for a Stackpole transit a 

 flexure-correction of the form (/cos z +f cos'^z) sec d. 

 But if the axis is reasonably stiff, the second term is 

 never sensible. — {Si(J. mess., June.) D. p. T. [1 



MATHEMATICS. 



Double theta-f unctions. — M. Caspary gives an 

 account of some of the more elementary theorems 

 concerning the theta-functions of two variables. He 

 proves first, in a very simple manner, that the squared 

 functions can he arranged in the form of a determi- 

 nant of the fourth order which satisfies all the condi- 

 tions of a determinant of an orthogonal substitution. 

 He derives also the G-opel relations between these 

 functions and their application to Kummer's surface. 

 A number of other fundamental theorems are also 

 arrived at in a very elementary manner, making the 

 paper a valuable introduction to the study of the 

 double theta-functions. — (Journ. reine ang. math., 

 xeiv. no. 1.) t. c. [2 



Periodic functions. — M. Hurwitz discusses sin- 

 gle-valued 2 n-fold periodic functions which through- 

 out a finite region have the character of rational 

 functions, and which are real for real values of their 

 arguments. More exactly he examines the properties 

 of the periods of such Abelian integrals as belong to a 

 real algebraic form (gehilde). By a real algebraic 

 form he means the aggregate of all pairs of values of 

 (x, y) which satisfy an irreducible algebraic equation 

 {F(x, y) = 0) whose coefficients are all I'eal. Defin- 

 ing a periodic function by the equation 

 ^(Mi + P,,u.i+ P.2. ■ ■ Jt,, + Pu) — <Hu,, U2 . . . M,,), 

 the complex of quantities Pa is called a period of the 

 function ip, a single one of these quantities being 

 called a modulus of periodicity. A period is then 

 real or pure imaginary when the moduli of periodicity 

 which constitute it are real or pure imaginary. The 

 principal theorem arrived at by the author is as 

 follows: let ifluu u-z . . ■ iin) denote a single-valued 

 2n-fold periodic function which everywhere through- 

 out a finite region possesses the character of a rational 

 function, and which takes real values whenever its 

 arguments are real ; then there are always n period- 

 pairs, 



(^"13,^23 • • • -Pjis); {Pi,n + p,P2,n + p ■ • • Pn,n+^)j 

 which form together a system of primitive periods of 

 the function, and which are of such a nature, that, for 

 each pair, one of the two conditions following is satis- 

 fied: either the first period {Pi^ . . . Pn^) is real, and 

 the second period (Pi, n + ? • • ■ Pn, n + p) is purely 

 imaginary ; or the first period is real, and the period 

 (2Pi,n-l-3 - -Pi.3- • • 2P,i, n + ? — Pnfi) is purely 

 imaginary. — {Journ. reine ang. math., xciv. no. 1. ) 

 T. c. [3 



PHYSICS. 



Iiiqnef action of nitrogen and carbonic oxide. 

 — S. Wroblewsky and K. Olszewski give a more 

 detailed account of the liquefaction of nitrogen 

 (Science, i. 970). The gas remained invisible when 



submitted to a pressure of one hundred and fifty at- 

 mospheres and a temperature of — 1.36° ; but, when 

 the pressure was slowly reduced to fifty atmospheres, 

 the gas was liquefied, presenting a visible meniscus, 

 and evaporating very rapidly. Under the same con- 

 ditions, the authors succeeded in liquefying carbonic 

 oxide, which formed a colorless liquid with a visible 

 meniscus. — [Comptes rendus. xcvi. 1225.) c. r. m. 



[4 

 Optics. 



Mirage. — In an article entitled " State of the 

 atmosphere which produces the forms of mirage ob- 

 served by Vince and by Scoresby," Prof. Tait presents 

 some very interesting researches regarding these par- 

 ticular forms of mirage. After an historical note, in 

 which he refers to two valuable contributions to the 

 subject by Wallaston and by Biot, which go far to- 

 ward a solution of the problem, but have unfortu- 

 nately fallen into oblivion, he presents his own 

 investigations. His method consists in treating the 

 curvature of a ray of light in the same way as the 

 motion of a projectile; the two cases corresponding 

 when, in the case of mirage, the square of the index 

 of refraction of the air is proportional to the distance 

 from a given horizontal plane. 



He finds, however, that, "whatever be the law of 

 refractive Index of the air (provided it be the same 

 at the same elevation), all we have to do to find the 

 various possible images of an object at the same 

 level as the eye is to draw the curve of vertices for 

 all rays passing through the eye in the vertical plane 

 containing the eye and the object, and find its inter- 

 section with the vertical line midway between the eye 

 and the object." By making suitable suppositions 

 regarding the change of density, he finds that " the 

 conditions requisite for the production of Vince's 

 phenomenon are a stratum in which the , refractive 

 index diminishes upwards to a nearly stationary' state, 

 and below it a stratum in which the upward diminu- 

 tion is either less, or vanishes altogether." It will be 

 seen that the solution of the problem of atmospheric 

 density and refraction by this means is entirely in- 

 determinate. The supposition of Prof. Tait satisfies 

 the conditions presented by the observations of Vince ; 

 but that it is the only law or the true law must be 

 verified by investigations of a different nature. 



The method is especially valuable in its inverse form 

 as affording a test of supposed laws of density and re- 

 fraction in their ability to furnish the various phe- 

 nomena of mirage. — {Nature, May 24.) g. e. c. 



[5 

 (^Photography.) 



Concentrated developer in one solution. — 



Where the photographer intends to travel, and devel- 

 ope on the route, it is very desirable to reduce his 

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 liquids possible. Mr. G. Cramer, the dry-plate manu- 

 facturer, gives the following formula for a developer, 

 which he considers gives the best of results, and at 

 the same time has the advantage of extreme porta- 

 bility. 



Stock solution. 



Sulphite of soda (crystals) .3 ounces. 



Bromide of ammonium i ounce. 



