28 



NATURE 



[May 14, 1908 



would have to be used with great caution in estimat- 

 ing tramwav requirements in Europe. Fortunately 

 there is no need to use American figures at all, since 

 sufficient data are available from European experi- 

 ence. A curve on p. 14 is interesting as showing 

 that with the expansion of towns the mileage of elec- 

 tric lines per 1000 inhabitants goes down, and the 

 vearly number of journeys made by each inhabitant 

 goes up. The figures are not directly applicable to 

 European towns, but the tendency shown by these 

 curves is the same in Europe. Towns of about 40,000 

 inhabitants show the greatest mileage, namely o'76 

 per 1000 inhabitants, but only no journeys per in- 

 habitant yearly, whilst towns of one million inhabi- 

 tants and above have on the average only half a mile 

 of line per 1000, but each inhabitant uses the cars on 

 an average 230 times a year. 



Jt is not clear from the author's figures whether 

 they refer to what we should term tramways or 

 whether they include railwavs also ; the latter is prob- 

 able, for tables giving mileage, equipment, cost, and 

 earning of electrified main lines are mixed up with 

 the other statistics. The next three chapters are de- 

 voted to what the author calls " Electrical Features." 

 and deal with motor capacity and running diagrams. 

 Various methods for getting out these curves are 

 given, namely, Armstrong's, .Storer's, and Hutchin- 

 son's methods, the latter at some length. The treat- 

 ment is by no means lucid, formula; and coefficients 

 being introduced without explanation. Unless the 

 reader is a thorough expert in this subject (when he 

 needs no further instruction from the author) he wil? 

 make nothing of these chapters. 



Altogether the author's mathematics is not char- 

 acterised by exactitude. Thus, on a later page, when 

 he treats of converters, following (with due acknow- 

 ledgment) Mr. Hay's method for the determination 

 of the output, we find him calling a line like the 

 following 



iI,r + iI,.--4I„/"cos2('a-']±I„T„rsin(a-" 



an equation, without saying what it is equal to, and 

 omitting the differential da. It will also be noticed 

 that the third term should contain either the product 

 of two currents or the square of a current, so that 

 the expression is also wrong in the matter of dimen- 

 sion. A reader having Mr. Hay's book at hand will 

 perhaps be able to find his way through the author's 

 mathematics, but without such aid he had better skip 

 the part on p. it/5. 



The author seems to pin his faith to the system, 

 almost universal in America, of transmitting by three- 

 phase current and converting into continuous current 

 by means of rotary converters in substations. Motor 

 generators, direct working, or the use of boosting 

 batteries are not even mentioned. The important 

 matter of heating of transformers and means of cool- 

 ing is dealt with in less than two pages of general 

 remarks, but to make up for this we get plenty of 

 catalogue pictures of plant installed by the two lead- 

 ing American companies. Chapter ix., treating of 

 NO. 201 I, VOL. 78] 



insulating oils, is instructive. On p. 234 a curve is 

 given showing the enormous influence on the insulat- 

 ing property of the oil of even slight traces of 

 moisture, and the specification given on p. 239 should 

 prove useful. Gisbert K.^pp. 



OXJR BOOK SHELF. 

 (i) Algebraic Equations. By G. B. Mathews, F.R.S. 

 Pp. viii-!-64. (2) Tlie Theory of Optical Instru- 

 ments. By E. T. Whittaker. Pp. viii-l-72. Cam- 

 bridge Mathematical Tracts, Nos. 6 and 7. (Cam- 

 bridge : The University Press, 1907.) Price 2S. 6d. 

 each net. 

 (i) The solution of a given equation is a problem 

 which has attracted the attention of many of the 

 greatest mathematicians. In this tract we have a short 

 summary of the results arrived at. The solution de- 

 pends on the properties of a certain permutation-group 

 called the Galoisian group; if this group is soluble, the 

 equation is solvable by radicals. Interesting tyoes oi 

 soluble groups are cyclical, .Abelian, and metacyclic 

 groups. To each of the corresponding equations is 

 devoted a chapter in which are explained the appli- 

 cation of cyclical groups to cyclotomy, the dependence 

 of .'\belian on cyclical equations, and Kronocker's 

 solution of the metacyclic equation. Prof. Mathews's 

 masterly epitome of the subject is not very easy read- 

 ing, and he assumes some knowledge of Tschirn- 

 hausen's transformation, the theory of permutation- 

 groups, &c. The student will probably have to pre- 

 pare himself for the study of this tract by reading 

 some more elementary treatise on the same subject 

 (e.g. Dickson's "Algebraic Equations"), and some 

 book on groups, such as Burnside's. 



(2) Dr. Whittaker does not follow Prof. Mathews 

 in writing for the advanced mathematician, but 

 appeals in the first place to those students of physics 

 to whom mathematics is interesting chiefly for its 

 applications. The professed object is to give " a 

 simple theoretical account of those defects of per- 

 formance of optical instruments to which the names 

 of coma, curvature of field, astigmatism, distortion, 

 secondary spectrum, want of resolving power, &c., 

 are given." Limitations of space necessitate in 

 places proofs which, though clear, are rather too con- 

 cise; but except for this the beginner will find the 

 tract fairly straightforward reading. The author has 

 succeeded in producing a book which will prove re- 

 markably interesting, not only to the user of optical 

 instruments, but also to any student of mathematics. 

 The leading principles and results are verv attrac- 

 tively presented, and can be readily grasped without 

 plodding through every detail of the somewhat 

 laborious approximations which the subject at times 

 requires. H. H. 



Detection of the Common Food Adulterants. By 

 E. M. Bruce. Pp. viiH-<S4. (London : A. Con- 

 stable and Co., Ltd., 1907.) Price 55. net. 

 The L'nited States used popularly to be looked upon 

 as par excellence the land of wooden nutmegs and 

 similar examples of perverted manufacturing in- 

 genuity. Perhaps, therefore, it is fitting that what 

 our author calls " the great pure food reform " should 

 find especial favour there. Be that as it may, there 

 has undoubtedly arisen in the .States a quickening of 

 interest in the matter of food adulteration ; wherefore 

 Mr. Bruce speaks of health officers, food inspectors, 

 chemistry teachers, and even students being con- 

 stantly called upon to test the purity of various foods 

 — at whose instance is not quite clear. He proposes 

 to help them and others in this task, which he says 



