586 



SCIENCE. 



[N. S. Vol. IX. No. 225. 



ter for the employer to be liberal in esti- 

 mating the time-rate rather than with the 

 premium-rate. Excessive premium-rates 

 are apt to result in too large expectations 

 to be fully met in the long run. From one- 

 half to one-third the saving are usual 

 premium- rates, and probably one-third to 

 the workman and two-thirds to the firm 

 best brings out a permanent and satisfac- 

 tory adjustment which, if found inequitable, 

 can generally be easily readjusted to a cor- 

 rect figure. In one machine-tool works 

 the premium-rate is thirty-six per cent, and 

 is found satisfactory to both sides. The 

 higher premium-rates, however, should be 

 paid for manual labor, as in blacksmithing, 

 and the lower to power- tool work, as at the 

 lathe or the planer or the milling machine. 

 "Undoubtedly every establishment, and 

 every department of labor, from floor- 

 sweeping to book-keeping, has its own 

 peculiar best rate. In all cases the result 

 may be expected to be a largely increased 

 output of the works, a greatly increased 

 earning power on the part of the men, and 

 decreased costs of production with increased 

 dividend-paying power for the holders of 

 the capital. "Wisely administered, the 

 plan will do more to settle the wages- ques- 

 tion than anything else that has been sug- 

 gested," and the wages-question is to-day 

 the burning question in the economics of 

 manufacturing. 



E.. H Thueston. 



SCIENTIFIC BOOKS. 

 Analytic Functions. Introductiou to the Theory 



of Analytic Functions. By J. Harknbss 



and F. Mobley. London, Macmillan & Co. 



1898 8vo. Pp. xvi + 336. 



The appearance of the present work is a very 

 pleasant sign to friends of the modern school of 

 mathematics in England and America. It in- 

 dicates that the movement which set in some 

 years past with us in this direction has been 

 steadily growing ; that the theory of functions 

 is no longer the property of a few bold and rest- 



less minds, but has ah'eady descended to the 

 masses. The present work may very happily 

 serve as a text or reference book to a first 

 course on the theory of functions in the senior 

 class of any of our better universities. The 

 theory of functions of a complex variable may 

 he viewed from two standpoints. One was 

 taken by Cauchy and Riemann ; the other by 

 Weierstrass. The methods of Cauchy and Rie- 

 man are more natural and intuitive ; those of 

 Weierstrass more abstract and lend themselves 

 more easily to a rigorous treatment of the sub- 

 ject. The authors have chosen the methods of 

 Weierstrass. 



Roughly speaking, the subjects treated in the 

 first 100 pages fall under two heads : 



1. The geometric representation of complex 

 numbers, the conformal representation afforded 



by 



ax-\-b 



^ ~ ex + d 



and the first properties of rational functions. 



2. Topics which lie at the foundation of the 

 calculus. 



The treatment of the first group of subjects 

 is admirable. In regard to the second is 

 seems^to us that the authors have attempted 

 the impossible. The theory of function in com- 

 mon with the calculus rests on certain notions, 

 such as that of number, limit, continuity, ex- 

 tremes of functions, etc. These subjects are 

 very imi^erfectly treated in Enghsh works on 

 the calculus, and our authors have thus found it 

 advisable to give some account of them in the 

 present volume. The amount of space at their 

 disposal was very limited, and they have, there- 

 fore, been obliged to be excessively concise. 

 This has been carried to such an extent in the 

 chapter on number. Chapter I., that the sub- 

 ject, so it seems to us, will be utterly incom- 

 prehensible to the student. 



We cannot understand why, if it is worth 

 while to say anything about irrational numbers, 

 the arithmetical operations upon them are passed 

 over in absolute silence. Until the terms sum, 

 product, etc. , are defined they have no meaning. 



Chapter VI., which treats of limits and con- 

 tinuity, suffers severely on account of the 

 brevity of Chapter I. In this chapter it is im- 

 portant to establish the existence of certain 



