April 27, 1883.] 



SCIENCE. 



337 



day when the advance of American geology' 

 seemed to depend on state surveys is passing, 

 and will soon pass a,via,y. Thej' did good 

 skirmish-work, and deserve to be remembered 

 for many gifts to science ; but the problems in 

 scientific geology are now too large to be solved 

 within the limits of a state. Scarce a state in 

 this country' has a question that can be proper- 

 \j considered from work done within its limits 

 alone. In the future the state survej's can find 

 their best place, not in efforts to develop gen- 

 eral scientific pi'oblems, but rather in economic 

 questions, wMeh can alwaj's be localized, and 

 in the work of bringing to the notice of the 

 people whom the^' serve such matters of pure 

 science as may naturallj'' concern them. Other 

 forms of research would better be left to the 

 general go^'ernment surveys, or to the studies 

 of independent geologists. 



It is now prettj' well ascertained that our 

 states are unwilling to support permanent sci- 

 entific establishments on such a scale as will 

 •enable them to do good scientific work, but 

 thej'' will pay some one or two men to keep a 

 sharp lookout for any utilities that may be 

 discovered. Fortunately nature so mingles 

 the ' utile ' and the ' dulce,' that some good to 

 science will come out of this care for profit, 

 which is to be in the future the task of the 

 state survej'or. 



M. HERMITE'S LECTURES. 



Cours de M. He>~mite, jyrofesse pendant le S' seinestre 

 1S81-82. liedige par M. Anoyer, e'leve de I'Ecole 

 normale supe'rieure ■ Second lirage revu par M. 

 Hermitk {Librarie scientifique) . Paris, A. Her- 

 mann, 1883. 



This work of M. Hermite fills, in great part, 

 a decided gap in mathematical literature, and 

 affords a means to American mathematical 

 students, at least, of overcoming a difHculty 

 that of late has become rather serious. With 

 the exception of those who have had the op- 

 portunity of listening to the lectures of Her- 

 mite or Weierstrass on the theory of functions 

 of a complex variable, all students interested 

 in that subject must have experienced a great 

 deal of difflculty in reading the more modern 

 memoirs which deal with it. Some such book 

 as Dur6ge's, or Neumann's, on Riemann's 

 theory, is verj' much wanted on what maj', with 

 propriet_y, be called the Weierstrass-Hermite 

 theorj' of functions. The necessity for such a 

 treatise is steadilj' increasing, as anj' one will 

 readily see by looking over the last few vol- 

 umes of Grdle-Borchardt, the Mathematische 

 annalen, the Annali di matematica, or the 



two numbers which have alread3' appeared of 

 Mittag-Leffler's acta mathematica. The pres- 

 ent work by M. Hermite does not profess to be 

 such a treatise. In fact, it is not a treatise at 

 all, but, as its title implies, simpl}' the course 

 of lectures given at the Sorbonne b}' M. Her- 

 mite, and treating of quite an extended list of 

 subjects. The principal topics discussed are 

 the quadrature and rectification of curves, the 

 determination of the areas and volumes of 

 curved surface, the general theory of functions 

 of a complex variable, and the application of 

 this theorj' to the study of the Eulerian inte- 

 grals and the elliptic functions. 



The first five chapters are devoted to geome- 

 trj^, and contain applications which are chosen 

 with a view to what is contained in the suc- 

 ceeding chapters. Since, for the rectification 

 of conies and the quadrature of plane cubies, 

 it is necessary to consider integrals of the form 

 / / (xy) dx, where / {xy) is a rational func- 

 tion of X and y, and y is the square root of a 

 quartic function of a;, the author takes up this 

 general integral, and gives Legendre's reduc- 

 tion to the normal forms of the elliptic inte- 

 grals, and also some of Tchebj-chef's results 

 concerning the cases where the elliptic inte- 

 grals are reducible to algeljraico-logarithmic 

 functions. 



The next three chapters are taken up with 

 an exposition of the more elementary properties 

 of functions of a complex variable, the author 

 giving an account of Darboux's investigations 

 relatively to the integral /* F (.v) / {x) dx, where 

 F (x) is, between the limits, always positive, 

 / (ft') is a continuous fauctiou of the form 

 cji {x) -\-i f (*') ; ^ncl where a and b are real. 

 Another method, due to Weierstrass, for inte- 

 grals of this nature, is also indicated. 



In the next four chapters the immediate 

 consequences of Cauchj^'s theorem are devel- 

 oped, and an accoun* given of Weierstrass's 

 and Mittag-LefHer's investigations in the theorj^ 

 of uniform fuuctions,. including their decompo- 

 sition of a holomorphic function into prime 

 factors, and their general expression for a uni- 

 form function with an infinite number of poles, 

 or of essential singular points, the last being 

 due almost solely to Mittag-LefBer. 



The next three chapters deal with the Eule- 

 rian integrals, and include Prym's expression 

 for r (x), and Weierstrass's expression for 



=— -r-1 and a demonstration by M. Hermite 

 T{xy ^ 



of Laplace's formula for the approximate cal- 

 culation of r (a;), where a; is a very large 

 integer. 



