310 THIRD REPORT — 1833. 



x'* — p x^ + q x^ — r X + s = 0, (1.) 



sixch functions would be x^ x^ + x^ x^ and (ic, + x^ — x^ — x^^, 

 which admit but of three different values, and which may seve- 

 rally form, therefore, the roots of cubic equations, whose coeffi- 

 cients are expressible in terms of the coefficients of the original 

 equation. Such a function also would be {x-^ + x^^, if we should 

 suppose p or the coefficient of the second term of equation (1.) 

 to be zero *. The function (Xj + ^2) (■'^3 + ^4) would give 

 three values only under all circumstances. The functions x^ 

 + ^2 + ^3 ^^^ ^1 ^2 •''^3 ^^^ capable of four different values, 

 and therefore do not admit of being expressed by a determina- 

 ble equation of lower dimensions than the primitive equation. 

 Functions of the form x^ Xo admit of six values, and require for 

 their expression equations of six dimensions, which are reduci- 

 ble to three, in consequence of being quasi recurring equations -j-. 

 Innumerable functions may be formed which admit of 12 and of 

 24 values, and one alternate function which admits of two values 

 only J. 



The success of such transformations in reducing the dimen- 

 sions of the equation to be solved, would naturally direct us to 

 the research of similar functions of the roots of higher equa- 

 tions than the fourth, which admit of values whose number is 

 inferior to the dimensions of the equation. We may presume 

 that, if such functions exist, they are rational functions, for 

 if not, their irrationality/ would increase the dimensions of 

 the reducing equation, and would tend to distribute its roots 

 into cyclical periods ; and what is more, it has been very 

 clearly proved that if equations admit of algebraical solution, 

 all the algebraical functions which are jointly or separately in- 

 volved in the expression of their roots, will be equal to rational 



* The first of these transformations involves the principles of Ferrari's, some- 

 times called Waring's, solution of biquadratic equations ; the second that of 

 Euler ; and the third that of Des Cartes. See the third chapter of Meyer 

 Hirsch's Sammhmg von Aiifgaben aus der Theorie der algehraischen Gleickuyigen, 

 which contains the most complete collection of formulae and of propositions 

 relating to symmetrical and other functions of the roots of equations with 

 which I am acquainted. The combinatory analysis receives its most advan- 

 tageous and immediate applications in investigations connected with the 

 theory of such functions. See also Peacock's Algebra, note, p. 6I9. 



■f The form of its roots being u and — , they are reducible by the same me- 

 thods as are applied to recurring equations. 



I See Cauchy, Cokvs d' Analyse, chap. iii. and noteiv. The use of such al- 

 ternate functions in the elimination of the several unknown quantities from n 

 simultaneous equations of the first order, involving « unknov/n quantities, 

 will be noticed hereafter. 



