312 THIRD REPORT 1883. 



pose is exceedingly difficult to follow, being unnecessarily en- 

 cumbered with vast multitudes of forms of combination, and 

 requiring a very tedious and minute examination of different 

 classes of cases ; and it was, perhaps, as much owing to the 

 necessary obscurity of this very difficult inquiry as to any im- 

 perfection in the demonstration itself, that doubts were ex- 

 pressed of its correctness by Malfatti* and other contemporary 

 writers. The subject, however, has been resumed by Cauchy in 

 the tenth volume of the Journal de VEcole Pohjtechnique , who 

 has fully and clearly demonstrated the following proposition, 

 which is somewhat more general than that of Ruffini : " That 

 the number of different values of any rational function of n 

 quantities, is a submultiple of 1 . 2 . 3 . , . n, and cannot be re- 

 duced below the greatest prime number contained in n, without 

 becoming equal to 2 or to 1." If we grant, therefore, the truth 

 of this proposition, it will be in vain to seek for the reduction 

 of equations of higher dimensions than the fourth, by transfor- 

 mations dependent upon rational functions of the roots. 



The establishment of this proposition forms an epoch in the 

 history of the progress of our knowledge of the theory of equa- 

 tions, in as much as it so greatly limits the objects of research 

 in attempts to discover the general methods for their solution. 

 And if the demonstration of Abel should be likewise admitted, 

 there would be an end of any further prosecution of such in- 

 quiries, at least with the views with which they are commonly 

 undertaken. 



Lagrange, in liis incomparable analysis of the different me- 

 thods which have been proposed for the solution of biquadratic 

 and higher equations, has shown their common relation to each 

 other, and that they all of them equally tend to the formation of 

 a reducing equation, whose root is 



Xi + u jTg + «^ .Tg + 0,^X4+ &c. 



where x\, x^, x^, &c., are the roots of the primitive equation, 

 and where « is a root of the equation 



a"-' + «»-2 ^ ^«-3 -f . . . a + 1 = 0, 



where n expresses the dimensions of the equation to be solved. 

 He then reverses the inquiry, and assuming this form as 

 correctly representing the root of the reducing equation, he 

 seeks to determine its dimensions. The beautiful process which 

 he has employed for this purpose is so well known f that it is 

 quite unnecessary to describe it in this place ; and the result, 



* Mcmnrie della Sor. Hal., torn. xi. 



t Resolution des Eqiiafions Ntimeriques, Note xiii. 



