May 9, 1889 



NATURE 



Z7 





non-homogeneous substitutions. And there must likewise 

 exist homogeneous integral functions of F(ji, z.^ of the Nth 

 degree in the two variables z^, z.^, containing one arbitrary 

 parameter, which remain unaltered, to a factor pres, for 

 each of the group of 2N homogeneous polyhedral substitu- 

 tions. The functions of the latter kind (" ground forms ") 

 are first determined. First, a special function Fj (of 

 degree N/vj is determined, such that Fj = o gives one 

 of the vi summits on the sphere, and the N/i*! — i 

 other points into which it is transformed by the poly- 

 hedral rotation?. Then F/i is a particular function 

 having the property sought for. By an elegant applica- 

 tion of the theory of invariants, two other functions, 

 F2 and F3, corresponding respectively to the v^ and v^ 

 summits, are derived from Fj. We then have for the 

 general invariant function required — 



AjFi"! + A0F./2 ^^■^■^■i, 



which contains only one arbitrary parameter, since there 

 is in each case an identity of the form — 



XlC'Ei"! + Aj'^'F/i + A3<^)F3''3 = o, 



connecting Fi, Fg, F3. In the particular case of the 

 ikosahedron we have — 



and Fi^ + Y^ - \12%Y^=:0. 



The non-homogeneous function Z is next discussed. It 

 is shown that any form of Z, whatever, is a linear rational 

 function, (aZ' +/3)/(7Z' -f 8) say, of any particular form 

 Z^ ; so that it is sufficient to determine a special form Z 

 subject to the conditions that Z assumes the values 1,0, cc, 

 for the v,, Vo, 1/3 summits respectively. It is found that 

 Z = cY»^-ii¥.^z, and thus the synthesis of the polyhedral 

 functions is completed. 



In Chapters III. and IV. of his first part, Klein dis- 

 cusses the inversion of the polyhedral functions. If, in the 

 polyhedral equation cY^y-iiY^'A ■= Z, we suppose Z given 

 and z required, z appears as an N-valued function of Z, 

 whose properties it becomes our business to discuss. 

 Parallel to this problem we have a " form-problem." There 

 are for each of the five polyhedra a set of three forms 

 which are absolutely invariant for the 2N homogeneous 

 polyhedral substitutions ; thus, for the ikosahedron, these 

 are the special forms Fj, Fo, F3, themselves. We may 

 then suppose the values of these absolutely invariant 

 forms given, subject to the identical relation which in all 

 cases connects them ; and require the values of Zj and Zj. 

 There are in each case 2N solutions of this " form-prob- 

 lem," and it is shown that these can all be obtained from 

 the N solutions of the corresponding polyhedral equation 

 by adjoining an accessory square root. It is, of course, 

 obvious that the N solutions of the polyhedral equation 

 and the 2N solutions of the corresponding form-problem 

 can all be derived from any one of them by the N non- 

 homogeneous and the 2N homogeneous polyhedral 

 substitutions respectively. 



A brief graphical discussion is given of the functions Z, 

 Zj, Z2 ; and it is shown that Z satisfies the differential 

 equation — 



Z' 2V Z'/ 2V(Z - I)-' 2v^''£' 



Z-i 



llv{- + I/Vg- - \lv{- - I 



2(Z - i)Z • • • ^^^ 



the left-hand side of which is the differential invariant 

 which Cayley has called the Schwarzian derivative of z 



with respect to Z (see Forsyth's " Differential Equations," 

 § 61 and chapter vi.) ; and that Zi and Z2 each satisfy the 

 linear differential equation— 



^ Z 4(Z - i)-L- \ 

 + Z 



^ 



+ I 



)-S}=°- 



Through the latter equations z-^ and z.-, are identified as 

 particular cases of the Riemannian P-function, and thus 

 connected with the theory of the hypergeometric series. 

 Here Klein's work comes in contact with the well-known 

 researches of Schwarz— " Ueber diejenigen Falle, in 

 welchen die Gaussische Hypergeometrische Reihe eine 

 Algebraische Function ihres vierten Eiementes darstellt " 

 {Crelle, Bd. 75). 



The inversion of the polyhedral functions is next con- 

 sidered from the standpoint of Galois's theory of the 

 resolution of an algebraical equation. An attractive out- 

 line of this theory is given, so far as it concerns the 

 problem on hand. The starting-point may be said to be 

 the famous theorem of Lagrange, which, in a generalized 



form, runs as follows: If R, Ri, Rj be rational 



functions of the variables .r„ jtj, . . . . .v«, and if R remain 

 unchanged by all the substitutions of the x's which leave 

 Rj, Rj simultaneously unaltered, then R can be ex- 

 pressed as a rational function of Rj, Rj, . . . . and of the 

 elementary symmetric functions of the -t^s. In particular, 

 if we characterize all the functions which admit {i.e. are 

 unaltered by) a given group of substitutions, G, as belong- 

 ing to the family G, we see that all the functions of any 

 family are rationally expressible in terms of one another. 



Suppose now that we have any algebraical equations, 

 f{x) = o, whose roots are x^, x^, .... x„, and we " adjoin " 

 thereto a group of asymmetric functions, Kj, Kg, . . . ., 

 of its roots, whose values along with the coefficients 

 of f{x) are supposed to be " known," then there exists 

 a group of substitutions, G, that, viz., for which Kj, 

 Ko, . . . . are unaltered, such that all functions of the 

 family G and no others are rationally expressible in terms of 

 the " known quantities." If R be a function of a-„ x^, .... 

 Xn. not belonging to the family, but say to the family g 

 where g is a sub-group in G of the order v = N/«^, then 

 we can form an equation tor R, viz., 



(R - Ri)(R - Ro) . . . . (R - R,.i) = o, 



whose coefficients are rational functions of the known 

 quantities. Such an equation is called a resolvent of 

 f{x) = o. All the resolvents constructed by means of 

 functions R which belong to the same family gx are 

 identical in the sense that they are rationally transform- 

 able into each other, and with these are also identical all 

 resolvents arising from functions belonging to the families 

 g», .0-3, . . . ., where ^2, ^3, are the sub-groups " associa- 

 ted with ^1 " in the main group G of the original equation. 

 There are therefore as many different kinds of resolvents, 

 as there are different sets of associated sub-groups in G. 

 The group r of every resolvent is isomorphous with the 

 original group G ; that is to say, we can order the two 

 groups so that to every substitution S in G corresponds one 

 S' in r, and to every combination of substitutions STU 

 in G corresponds SiT^U^ in r. If this corre- 

 spondence be holohedric (one S for every S^), then the 

 resolvent and the original equation are equivalent in the 

 sense that every root of the one is rationally expressible 

 in terms of the roots of the other and of the known quan- 

 tities ; each is in fact a resolvent of the other. Pre-eminent 

 among this species of resolvents stands the Galois 

 resolvent, whose R is a perfectly asymmetric function as 

 regards the substitutions G. The degree of this resolvent 

 is the highest possible, viz. N. Since the sub-group ^1 

 belonging to any root Ri of this equation reduces to the 

 identical substitution, it follows that we can express each 



