338 



SCIENCE. 



[Vol. I., No. 12. 



The next two chapters I'efer to fiiuctions 

 which are discontinuous along a line, — Ap- 

 pell's and Tannery's series, and Poincarrd's 

 example of a function having an espace lacu- 

 naire. As prelirainarj' to Cauch3''s theorem 

 concerning the number of roots of a polyno- 

 mial contained in the interior of a contour, the 

 expression is given by a line-integral of roots 

 of an equation contained within a given con- 

 tour. Then follows Cauchy's theorem, the 

 establishmentof Lagrange's series, Eisenstein's 

 theorem upon series whose co-efflcients are com- 

 mensurable, and which satisfy an algebraical 

 equation, and the enunciation of Tchebychef 's 

 theorem upon series with rational co-efflcients, 

 which may represent functions composed of 

 algebraic, logarithmic, and exponential func- 

 tions. 



The next chapter treats of multiform func- 

 tions arising from the integration of uniform 

 and of multifoi'm functions, and of the means 

 of reducing them to uniform functions by sys- 

 tems of cuts (coupures) . 



The remaining five chapters treat entirely 

 of the doubly-periodic functions. After first 

 showing the multiple values of the elliptic in- 

 tegrals of the first kind which correspond to 

 the different paths traced out by the variable, 

 and establishing the double periodicity of the 

 inverse functions to this integral, he defines a 

 function, $ (x) , which conducts to the analyti- 

 cal expressions for the doublj^-periodic func-' 

 tions. The function ^ (x) is defined by the 

 equations, — 



^{x + a) = $(a;) 



*(» + b) = *(«) exp.f- '^''^ {2x -H 6)] , 



where k is an integer. Tlien follows the in- 

 vestigation of the elliptic functions, including, 

 of course, Jacobi's ®, H, and Z functions, the 

 definition of Weierstrass's functions, Appell's 

 expression for doublj'-periodic uniform func- 

 tions in the case where they possess essential 

 singular points, and, finall,y, a demonstration 

 by M. Goursat of Fuch's theorem concerning 

 the definite integrals K and K', considered as 

 functions of the modulus. 



It is perhaps to l)e somewhat regretted that 

 the book is lithographed instead of printed in 

 the usual manner ; but this is of comparativel_y 

 little consequence, as the writing is very clear 

 and legible. Thanks are certainly due to M. 

 Andoyer, the editor, for the trouble which he 

 must have taken in elaborating what would 

 seem to have been merely a set of notes on 

 M. Hermite's lectures. The whole matter has^ 

 been revised by M. Hermite, and the aggre- 

 gate result of his and M. Andoyer's labors is 

 a book which is a decided acquisition to mathe- 

 matical literature. It is to be hoped that M. 

 Hermite will see fit to go more fully into the 

 subject of the functions of a complex variable, 

 and that of elliptic functions, at a future time, 

 and give to the world a treatise which will be 

 more satisfactory than even the present very 

 valuable work. T. CRAifi. 



WEEKLY SUMMARY OF THE PBOG-BESS OF SCIENCE. 



ASTRONOMY. 



NeTV measures of Saturn's rings. — O. Striive 

 gives the results of a series of measurements of the 

 rings of Saturn at Pulkowa Quring August and Sep- 

 tember, 1882, compared with a similar series, also 

 taken by himself, with the same Instrument, and at 

 the same time of the year in 1851. In a memoir on 

 the subject In 1851, he seeks to prove, that, while the 

 outer diameter of the rings remains constant, the 

 inner is contiiuially shortening, basing his conclu- 

 sions on the observations and drawings from Huy- 

 gens's time. If the conclusion were correct, and the 

 contraction constant, the measures of 1882 should 

 have given a perceptibly shorter inner diameter than 

 those of 1851. The Inner diameter of the dark ring 

 seems to be slightly shorter than in 1851, but the 

 difference Is not nearly so large as the theory calls 

 for. The dark ring seems, however, to have changed 

 since 1851. Then It seemed divided by a dark streak, 

 the inner part being entirely separate from the bright 

 ring. In 1882, all trace of this division had disap- 

 peared, and the dark ring seemed to be merely a faint 

 continuation of the hright ring. — {Astr. nadir., No. 

 2498.) M. McN. [688 



Formation of the tails of comets. — Mr. Rum- 

 ford suggests that the repulsive force which is un- 

 mistakably manifested in the formation of comets' 

 tails may be due, not to any electric action, or any 

 imagined impulse of solar radiations, but merely to 

 evaporation. A small particle from which evapora- 

 tion is taking place on the side next the sun will be 

 driven backward with a velocity continually acceler- 

 ated ; and, when more than half of the mass of 

 the particle has been evaporated, the velocity of the 

 residue may be much greater than the avei-age veloci- 

 ty with which the gaseous molecules are driven ofiE 

 from the heated body. In the case of hydrogen at a 

 temperature of 70° or 80° F., the velocity thus ac- 

 quired might be greater than a hundred thousand 

 miles a day. If we suppose the evaporating material 

 to be gases which have been liquefied by the cold of 

 space (carbon dioxide and volatile hydrocarbons), it 

 becomes easy to account for a powerful repulsive 

 action at distances from the sun even much greater 

 than that of the earth. The writer suggests that the 

 comet's light maybe in part due to the ' bombardment' 

 of precipitated particles by the evaporated molecules 

 in the condition called by Crookes ' the fourth state 

 of matter'; so that, "without electrical discharges, 



