38 



NA TV RE 



[February 9, 1893 



"nebulosity," "nebulous light," and "nebulous matter," 

 when he means luminosity and luminous matter. In 

 ante-spectroscopic days the terms nebula and cluster were 

 used almost indiscriminately, a nebula being looked upon 

 as simply an irresolvable cluster, and this error still 

 survives in many astronomical text-books and com- 

 pilations, but Dr. Boeddicker should have avoided it. 

 When we consider that the majority of the stars in the 

 cluster which we call the Milky Way are of the Sirian 

 type, we see how misleading is the use of the terms 

 nebulous light and nebulous matter. A. T. 



THE THEORY OF SUBSTITUTIONS AND ITS 



APPLICATIONS TO ALGEBRA. 

 The Theory of Substitutions and its Applications to 



Algebra. • By Dr. Eugen Netto. Translated by F. N. 



Cole, Ph.D. (Mich. ; Ann Arbor, 1892.) 



THE theory of substitutions abstractly considered is 

 concerned with the enumeration and classification 

 of the permutations of a set of n different letters 

 .t'j, X2, . . . ., Xn- It is scarcely apparent at first sight that 

 a far-reaching mathematical theory could be built on a 

 basis so simple, still less that there should be any con- 

 nection between this and the complicated question of the 

 solution of algebraical equations by means of radicals. 

 It may be worth while, in order to excite the interest of 

 mathematical readers in the work before us, to mention 

 one or two points in the Theory of Substitutions which 

 will give an inkhng of the nature of its connection with 

 the interesting problem just mentioned. 



The operation of replacing — say in any function 

 ^(^i» •^'2» ^'zl — ^"y permutation of the letters, say Xi,Xo,x^, 

 by any other, say x^, x^, x^, is called a substitution. This 



operation is denoted explicitly by ("^^i' ■^2> ^z\ or shortly 

 by a single letter s. Thus s(^{x-^y x^, x^) = <t>{xj^, x^, .r.^) ; 



(x^, X2, x^ 

 X.,. Xn. X-, 



and again : If t denote the substitution 



t(p{x-^, A-3, x.^ = (^(jTg, jTj, x^. We may indicate the sue. 

 cessive application of the two substitutions s and t by 

 multiplying the symbols st in the order of application : 

 thus J/0(xi, A'2, x^ = (fiixs, Xi, jTjj) and tscj^ix^, x.^, x^) = 

 4>{x2, X3, x^). In particular, the repetition of the same 

 substitution may be represented by powers of the symbol ; 

 thus s^(fi(xi. X2, X3) = (l){xi, X2, x^). The identical substi- 

 tution C"^!' ^2. ■*'3\ is represented by unity. The total 

 VjTj, X2, x^f 



number of different substitutions of n letters is obviously 

 n.^ ; consequently, if we form the consecutive powers of 

 any substitution we shall ultimately arrive at a power s'" 

 which will be the identical substitution, 7n being some 

 positive integer not exceeding n.' : 7n is called the order 

 and n the degree of the substitution- 



If among the substitutions of any given degree we can 

 select a set which have the property that the product of 

 any two furnishes another substitution belonging to the 

 set, we obtain what is called a group of substitutions. 

 The whole of the ti! substitutions of n letters obviously 

 form a group, and the identical substitution by itself forms 

 a group. It is easy, however, to see that in general there 

 are other groups among the substitutions of a given 

 NO. 1215, VOL. 47] 



degree. Consider, for example, any rational function 

 *^(-''i> •^"2> • • • •, •*'«) which is not wholly asymmetric : 

 there must exist a set of substitutions each of which 

 leaves the value of unaltered. A substitution which is 

 the product of any number of these must also leave ^ 

 unaltered : hence the set in question forms a group. We 

 have here a fundamental point in the theory of substitu- 

 tions, viz., the existence of a group of substitutions and 

 the correlation therewith of rational functions which are 

 unaltered by all the substitutions of the group. The 

 group is said to belong to all the functions which it leaves 

 unaltered ; and these functions are said to form a family 

 which is characterized by the group. Thus the group of 

 a wholly asymmetric function is the identical group con- 

 sisting of the substitution ^ ; the group of the wholly 

 symmetric functions consists of the whole of the n! sub- 

 stitutions of the «'h degree ; the group of the alternating 

 functions consists of all those substitutions which are 

 equivalent to an even number of transpositions, and so 

 on. It is obvious that every rational function determines 

 a group of substitutions, and it may be shown that, con- 

 versely, for every group of substitutions we may construct 

 an infinity of rational functions which are unaltered by 

 the substitutions of the group. The significance of this 

 correlation between a group and a family of functions 

 depends on the following important theorem, which is due 

 in substance to Lagrange. If >//■ be a rational function 

 which is unaltered by all the substitutions of the group of 

 (/) (in other words, if the group of -^ contain the group of 

 0) then ^ can be expressed as a rational function of ^, 

 and the ;/ elementary symmetric functions 



Cx == 2a-j, C2 = 2A-1, X2, . . . ., Cn = Xu X2 . . . . Xn. 



A particular case of this is the theorem that if the groups 

 of -^ and <^ be identical, then each can be expressed as a 

 rational function of the other, and of the elementary 

 symmetric functions. A limiting case of this theorem is 

 the familiar result that every rational symmetric function 

 can be expressed as a rational function of the elementary 

 symmetric functions. As a special example consider the 

 the two wholly asymmetric functions r//' = ax^ + bx2, 

 (f) = a/xi + b/x2 : these both belong to the identical 

 group, since they are changed by every substitution of 

 the letters Xj^, X2. Hence -^ can be rationally expressed 

 as a function of (f), Q, Cg. The actual expression is in 

 fact 

 V^ = {2{a-byC2 - («2 + b-^)Ci^ + (a + ^)CiC2(^l { - (^ -f ^)C, 



The application of the theory of substitutions is limited 

 in the first instance to rational functions. Its use in the 

 theory of the solution of algebraical equations by means 

 of radicals is based on the following important result in the 

 theory of irrational functions. Any root of a solvable equa- 

 tion /(or) =0 can be expressed as a rational integral func- 

 tion of certain elements V^, Vo, . . . ., V^, the coefficients of 

 which are rational functions of the coefficients oif{x) and 

 of primitive roots of unity. The quantities Fj, V2,,.. ., Vn 

 are on the one hand rational integral functions of the 

 roots of /(.i) = o and of primitive roots of unity, and on 

 the other hand are determined by a series of equations 



Vap. = Fa{Va.-l, Va-2, . . . ., V.), 



where /a is a prime number and F is a rational function 

 of the V—s. For example, in the case of the cubic 



