2l8 



NATURE 



[January 6, 1898 



two non-consecutive curves touch, or (.r, y) is a cusp or 

 point of self-contact of one particular curve (f){.v,jy, c) = o. 

 Thus we may have the envelope of the family of curves, 

 a tac-locus, or a locus of cusps or of points of self- 

 contact. On the other hand if we eliminate c from 

 {x, _y, c) = o and d(f)jdc = o, we get the locus of inter- 

 section of consecutive curves <p : this may include besides 

 the envelope proper, a locus of nodes, of cusps, or of 

 multiple points of higher order (as, for instance, points 

 of self-contact or triple points). The only outstanding 

 difficulty is the degree of multiplicity in which the 

 singular loci, distinct from the envelope, are involved in 

 the two discriminants. 



There are si.x papers on the transformation of elliptic 

 functions, the most important being No. 578. This 

 contains an exposition of the Jacobian theory, Sohnke's 

 modular equations with additions, and a discussion of 

 the singularities of some of the modular curves. It is 

 remarkable that Cayley, like Kronecker, adhered firmly 

 to Jacobian methods, and never seems to have worked 

 with the Weierstrassian forms. Perhaps just now there 

 is a rather exaggerated tendency in the other direction : 

 as Prof Klein has pointed out, both theories are self- 

 consistent and form, in a sense, the first and second 

 stages in a complete discussion of periodic functions. 



There is not very much about invariants and co- 

 variants ; No. 525 is an interesting example of a quad- 

 ratic transformation, and the papers on "trees," although 

 ostensibly intended for application to chemistry, were 

 suggested by the invariant calculus. 



In arithmetic there is a table of reduced binary cubics 

 with their Hessians, which is a development of Arndt's 

 results. Cayley gives the composition tables for the 

 Hessians. 



Volume ix. contains eleven papers dealing more or 

 less with astronomy and dynamics ; and it may be worth 

 while to notice that this volume also contains a reprint 

 of the British Association " Report on Mathematical 

 Tables." 



Many interesting special points suggest themselves to 

 the reader : thus, to mention only three, very different 

 in character, the very simple and pretty proof of 

 Vandermonde's theorem (viii. p. 465) might very well 

 find a place in an elementary text-book of algebra ; we 

 are told {ibid., p. 188) how a theoretical error was 

 detected by a numerical calculation ; and {ibid., p. 397) 

 there is an unverified conjecture that every surface of 

 negative deficiency may be derived by a rational trans- 

 formation from a cone whose deficiency is equal to that 

 of the surface with its sign changed. G, B. M. 



EXPERIMENTAL PHYSICS. 

 The Outlines of Physics. By Prof £. L. Nichols. 



Pp. xi -f 452. (London : Macmillan and Co., Ltd., 



1897.) 

 Lessons in Elementary Practical Physics. Vol. iii. Part i. 



Practical Acoustics. By C. L. Barnes. Pp. x -f 214. 



(London : Macmillan and Co., Ltd., 1897.) 



THE first of these books, as the author explains in his 

 preface, is an attempt to " outline a short course in 

 physics which shall be a fair equivalent for the year of 

 advanced mathematics now required for entrance to many 

 NO, I47I, VOL. 57] 



colleges -' ; and he proceeds to point out that if physics is 

 to possess much disciplinary value, it must be taught by 

 laboratory methods. Experimental work thus finds a 

 prominent place in his book, which may, in fact, be 

 roughly described as a series of experiments, mostly suit- 

 able for repetition by young students, connected by short 

 discussions of a theoretical character. 



With the author's object we imagine that most teachers 

 of physics will cordially sympathise. That experiment is 

 the means whereby a knowledge of physics should be 

 acquired by beginners, is as clear now-a-days as it is that 

 the means itself is open to improvement — at any rate, in 

 its early stages. Whether the author has made the most 

 of his opportunity is, however, less certain. Much of his 

 work is excellent : the experiments are, for the most part, 

 well chosen and clearly described ; but after a careful 

 perusal of his book, one's prevaihng impression is that he 

 has attempted to include too much. 



A book of this kind is, of course, largely taken up with 

 description of experimental procedure ; but the space is 

 often further occupied with matter which might, in our 

 opinion, be left until a later stage in the student's career. 

 Such questions as X-rays, tests for and theory of colour- 

 blindness, interference and polarisation of light, are too 

 large for more than the briefest notice, and might there- 

 fore just as well have been omitted altogether ; especially 

 when, to mention one instance out of many, curved mirrors 

 are dismissed with a far too scanty discussion, and no 

 special experimental illustrations at all. It would, in our 

 opinion, have been better to develop further the experi- 

 mental treatment of the simpler parts of physics at the 

 expense of these more elaborate phenomena. It is only 

 in places, however, that the work is affected by this fault ; 

 and the same may be said of an occasional laxness of 

 expression which will probably lead to mistakes on the 

 part of young readers where it occurs. Taken as a 

 whole, the book forms a useful addition to the elementary 

 text-books on practical physics. 



We have noticed a few points that rather need altera- 

 tion. In the figure of the apparatus /or determining the 

 heat of vapourisation of water (p. 172), the long tube 

 connecting flask and calorimeter should be provided with 

 a trap for the steam condensed in it. The statement in 

 italics on p. 213, that " various bodies can be brought by 

 friction {i.e. by doing work upon them) into a condition 

 such that they attract and are attracted," is rather mis- 

 leading. It is, of course, the work done in pulling the 

 rubber and rubbed object apart which should be em- 

 phasised. On p. 337, in the figure illustrating the motions 

 of the air in sound waves, the arrows want altering ; on 

 pp. 308 and 310, misprints of iron for ion, and ammonium 

 for ammonia, respectively, occur ; and on p. 99, in the 

 last column the decimal point has gone astray. 



The general get-up of the book is, as one would ex- 

 pect, excellent ; and the diagrams, which are mostly by 

 Mrs. Nichols, are very clear and well executed. We may 

 add that the work is almost wholly non-mathematical. 



The second of the two books named at the head of this 

 notice, forms the first part of vol. iii. of the " Elementary 

 Practical Physics" series begun in 1885 by Prof Balfour 

 Stewart and Mr. W. W. Haldane Gee. 



With the rapid development of the teaching of physics 

 by laboratory methods, now in progress, has arisen the 



