March 9, 19 11] 



NATURE 



39 



support the alleged distinction between C. vulpina 



and its var. nemorosa, C. canescens, and its var. 



robiisttor, C. diandra and its var. Ehrhartiana ; shows 



some difference between C. Goodenowii and var. jun- 

 , cella, much more between C. binervis, Sm., and var. 



Sadler i, Linton, which are farther apart than has 



been supposed. 

 This posthumous work, which throws more light 



on the Carices than most of us expected, owes its in- 

 j ception and completion to its distinguished editor, but 

 ! it is a fine memorial of the persevering toil and ability 

 I of its lamented author, which must find its way into 

 : the hands of every botanist who pretends to a know- 

 I ledge of the genus. Edward F. Linton. 



I TABLES OF SYMMETRIC FUNCTIONS. 



The Symmetric Function Tables of the Fijteenthic, 



including an Historical Summary of Symmetric 



Functions as Relating to Symmetric Function 



Tables. By Prof. F. F. Decker. Pp. i6 + tables. 



I (Washington, D.C. : Carnegie Institution, 1910.) 



' nPHE publication of this paper is an example of 



A the excellent work tha*: is being done by the 



Carnegie Institution of Washington, over a wide field 



i of science, under the fostering care of Dr. R. S. 



Woodward. 

 'i The formation of Symmetric Function Tables dates 

 from the first decade of the nineteenth centurv, when 

 I Meyer Hirsch gave them up to and including the 

 loie. ^ These tables give the expression of a symmetric 

 function of the quantities, a^, a^, . . . a^^, in terms 

 of the elementary symmetric functions thereof, p^, 

 p2 • • ■ Pio> where 



{x-a^){x-a^) . . . (x-a,g)=x^°-/,^x^+ . . . +Ao- 

 According to modern notation and nomenclature a 

 ! function 



2ai''iff/2 _ _ _^^^,r]o or (ir^n^ . . . ir^^) 



is thus expressed in terms of quantities 



/, = 2aja2 ... a, or (l*), 

 the exponents of the quantities, a under the sign of 

 summation being merely assembled in a bracket. 



Ex.i;r. (22) = (i2)2-2{l)(l3) + 2(l^), 



and it will bo observed that each term on the eight 



involves four units, and each is said to be a separa- 



^•n of (I'V Mr. Decker's tables, like the earlier 



- to which he refers, express all the functions 



^n. . .), where 2^=15 in terms of separations of 



I, and a reader of his historical summary might 



.>unnose that nothing further had been done in the 



!way of .tables of symmetric functions. The facts are 



quite otherwise, for several remarkable extensions 



have been made to which Mr. Decker makes no 



■(reference whatever. So far back as 1888, in the 



^"^erican Journal of Mathematics, an analogous 



■ry was shown to exist in regard to the separations 



►r any partition whatever, and a complete set of 



jtables, direct and inverse, up to weight six inclusive, 



|vvas given in the Journal; a law of symmetry corre- 



•^ponding to the Cayley-Betti law and several other 



^ of symmetry were established. For example, 



row for weight six and the separable partition 



') is :— 



2(5I) = (3)(2I) + (2){V)-(I)(32)-(32I). 



NO. 2ik8. vol. 861 



It was also shown that the weight might be 2ero 

 or negative, and the separable partitions involve zero 

 and negative parts without interference with the con- 

 struction of the tables or with their fundamental 

 properties. 



Further, also in the Phil. Trans., R.S., 1890, the 

 tables and properties were extended to the sj^mmetric 

 functions of several systems of quantities and specimen 

 tables were given. It is necessary to say so much, as 

 otherwise a reader might be grievously misled. 



The historical summary in regard to a single system 

 of quantities and the separations of (i«) is well given 

 by Mr. Decker. He is in error in ascribing formulas 

 for calculating the constituents in each of the first four 

 lines or columns to Roe; the first of these is nothing 

 more than Waring 's law, extended by the law of 

 symmetry; the others are readily obtainable from it, 

 and have long been in use by calculators. 



The chief use of the formation of tables of these 

 functions has been that the construction has led to 

 the discovery of new theorems which have been of 

 use in other departments of mathematics; in par- 

 ticular, remarkable differential operators were thus 

 brought to light which have been successful in open- 

 ing up problems of the magic square description, 

 which had defied analysts from Euler to Cayley. Also 

 the theory of non-unitary symmetric functions was 

 shown to involve that of the covariants of binarv 

 quantics. 



It is disappointing to find that Mr. Decker's 

 laborious work with the 151° has not resulted in the 

 discovery of theorems of wide application ; this is not 

 surprising, because it is fairly certain that previous 

 workers have taken the principal plums out of this 

 particular orchard ; but, this being so, there does not 

 appear to be any sufficient reason for continuing this 

 series of tables. 



Mr. Decker's table is beautifully produced, and he 

 seems, while detecting errors or misprints in the 

 lower tables, to have carried out a good system of 

 checks to ensure freedom from error in his own. 



P. A. M. 



WORKSHOP MATHEMATICS. 

 Shop Problems in Mathematics. By W. E. Breckcn- 

 ridge, S. F. Mersereau, and C. F. Moore. Pp. vii + 

 280. (Boston and London : Ginn and Co.) Price 

 45. 6d. 



THE authors' aims in producing this book have 

 been to impart to the student information in 

 regard to the more important points in shop work, 

 such as the names of the parts of machines used in 

 wood and metal working, together with the materials 

 employed, and also to give a thorough training in the 

 mathematical operations that are useful in shop prac- 

 tice and science. In carrying out these ideas, about 

 two-thirds of the space available are occupied in 

 calculations applied to timber, house building, 

 machines, pattern-making, and foundry work, forg- 

 ings, screws, and screw-cutting, and the gas engine. 

 The latter part of the book is taken up with a review 

 of calculation with short methods. 

 The book is decidedly American in its arrangement, 



