38 



NATURE 



{May 9, 1889 



of the x's in terms of Rj. In fact, all the roots of the 

 Galois resolvent are rational functions of any one of them ; 

 and it has the remarkable property of being transformable 

 into itself by a group of N rational transformations, which 

 stands in holohedric isomorphism with the group r. It 

 is also established that an irreducible equation of the Nth 

 degree which is transformable into itself by a group of N 

 rational transformations is its own Galois resolvent ; and 

 its group is holohedrically isomorphous with the group of 

 the N transformations. We are now enabled to perceive 

 a very important property of the polyhedral equations, 

 viz. each of them is its own Galois resolvent ; the N 

 rational transformations in question being simply the N 

 linear polyhedral transformations. Every polyhedral 

 equation therefore stands in a fundamentally simple rela- 

 tion to any equation of which it can be shown to be a 

 resolvent. 



If we consider the case where the isomorphism of the 

 groups G and r is not holohedric— that is, where to each 

 of the S^'s corresponds a group of the S's, we see that this 

 necessitates the existence of a self- conjugate (jZi^z'-asso- 

 ciate) group y in G to which belong the whole of the R's. 

 If to the other known quantities we now adjoin the R's, the 

 solution of the original equation fi^x) = o will be simpli- 

 fied, because its group is now y, which is smaller than G. 

 Moreover, the R's themselves are calculable in terms of 

 the known quantities by means of an equation whose 

 group r is also smaller than G. In this case, therefore, 

 an essential simplification in the formal solution of the 

 equation /(a-) =o can be effected. If the group y be 

 either intransitive or composite, a further simplification 

 would ensue, in the one case by the "reduction" of 

 /{x) = o, in the other by the construction of another 

 resolvent having a smaller group than y. 



The application of the latter part of the general theory 

 in combination with the data regarding the groups of 

 the polyhedral substitutions obtained in the earlier 

 chapters, leads at once to important conclusions re- 

 garding the polyhedral equations. It is found that the 

 octahedral equation can be solved by extracting in 

 succession a square root, a cube root, and finally two 

 square roots ; the tetrahedral equation by the same 

 series of operations, if we omit the first, and the di- 

 hedral equation by extracting a square root and then an 

 nth root. All these equations are therefore soluble by 

 means of the ordinary elementary algebraical irration- 

 alities. 



The ikosahedral equation stands by itself because the 

 ikosahedral substitutions form a " simple " group ; its 

 lowest resolvents correspond to the five associate tetra- 

 hedral and the six associate dihedral sub-groups of the 

 ikosahedral main group, and are of the fifth and sixth de- 

 grees respectively. This is, from one point of view, the 

 main part of the theory, for it leads us to see that the 

 solution of the ikosahedral equation involves an irra- 

 tionahty which exists independently of the ordinary 

 algebraical irrationalities. 



Since Abel demonstrated the impossibility of solving 

 general equations whose degree exceeds the fourth by 

 means of elementary algebraical irrationalities, two, 

 or perhaps we should say three, great classes of 

 problems in the higher theory of equations have arisen : 

 (i) to characterize and classify all those exceptional 

 cases of equations of a degree exceeding the fourth which 

 can be solved by elementary irrational operations ; 

 (2) to circumscribe the domain of the higher algebraic 

 irrationalities — that is, to characterize and, exhaustively 

 classify all the essentially distinct irrational operations 

 which are required for the solution of any algebraical 

 equation of finite order,— this is not to be confounded 

 with the interesting and practically important, but 

 perfectly distinct, question regarding the solution of such 

 equations by means of transcendental irrationalities, such 



as circular and elliptic functions ; (3) in connection with 

 each distinct higher irrationality, there arises, of course, 

 the question as to the characteristics of the various 

 equations which can be solved by means of this 

 irrationality and others of a lower order. 



Much has been done in the working out of the first 

 problem by Abel, Kronecker, and others ; but compara- 

 tively little progress has been made with the second 

 class of problems. In Klein's work we have an 

 important contribution to this new branch of the theory 

 of equations ; and a sketch of a general method which 

 seems to promise farther advance in the immediate future. 

 The latter part of the book under review is almost en- 

 tirely occupied with this subject. He there shows by two 

 different methods that the solution of the general quintic 

 equation can be effected by means of the ikosahedral 

 irrationality combined with an accessory square root. A 

 brief sketch of his first method will enable the reader to 

 understand the general march of the investigation. If to 

 the rational coefficients of the quintic equation we adjoin 

 the square root of its discriminant, its Galois group be- 

 comes the 60 even permutations of its roots. Now this 

 is isomorphous with the group of the ikosahedral equa- 

 tion, and therefore (since that group is simple) with the 

 group of any of its resolvents. But it is shown that one 

 of the ikosahedral resolvents (" the principal resolvent '') is 

 an equation of the fifth degree of the form' j/^ + SQ/'' -f 

 5R/-f- S, where O, R, S are rational functions of three 

 arbitrary parameters ot, «, Z. The question then naturally 

 arises. Can we rationally connect the roots of this re- 

 solvent with the roots of the general quintic by properly 

 determining the parameters ;«, «, Z ? By means of a 

 Tchirnhausian transformation, we can reduce the general 

 quintic to a " principal equation " of the formy -f ^oyj -\- 

 Sl^y + 7=0; and it is shown that the necessary operations 

 become rational after the adjunction of the square root 

 of the discriminant of the quintic. We have thus two 

 equations, each involving three arbitrary parameters ; an.d 

 it is shown that the determination of ;//, n, Z in terms of 

 a, /3, y so as to satisfy the equations 0=?a, R = ^, 8=7 

 involve no farther irrational operations. The calculations 

 in both methods are full of beautiful details, partly 

 geometrical and partly analytical in character. 



In the last chapter of the first part a general survey of 

 the theory of the polyhedral functions is given, wherein 

 their relation to a variety of other functions is pointed 

 out. In particular it is shown that the polyhedral 

 functions virtually embrace all functions that "admit" a 

 fijtite group of linear transformations. The proof of this 

 depends essentially on the fact that the diophantine equa- 

 tion, 2(1- i/i-z) = 2 — 2/N, where the vi's and N are 

 all finite and positive, has only four solutions, viz. the 

 values of i/j, v^, "3, N (given in the above table), which 

 characterize the polyhedral functions. In these four cases 

 i/"! + ^l^-i -f l/i's > I. If in the differential equation 

 (A) we give to Vj, i/j, v^ other integral values for which 

 i/i'i + i/j/g -f- i/i's = or < I, we get the Schvvarzian func- 

 tions, which are transcendents admitting infinite groups 

 of linear substitutions. Among these, as a limiting case 

 corresponding to v^ =2, ^2 = 3, I's = 00 , are found the 

 elliptic modular functions. This fact naturally leads to 

 the attempt to solve the polyhedral equations by means of 

 transcendentally irrational functions ; and it is shown 

 that, just as the binomial equation, Z'^ = A, can be 

 solved by means of logarithms, and the dihedral equa- 

 tion, z"-Yz--^= - 4Z, -f 2, by means of circular 

 functions, so the tetrahedral, octahedral, and ikosahedral 

 equations can be solved by means of elliptic modular 

 functions. 



The above imperfect sketch of Klein's " Ikosahedron '" 

 will, we trust, be held sufficient to justify us in saying that 

 Mr. Morris's hope that his translation " may contribute 

 towards supplying the pressing need of text-books upon 



