386 Prof. Cayley on the Solvibility of Equations 



fore probable that almost every trace of the ice of the Miocene 

 period would be removed before the glacial epoch began. From 

 the close of the glacial epoch of the Middle-Eocene period, repre- 

 sented in Table II., to the commencement of the glacial epoch of 

 the Miocene period, there is the enormous interval of 1,480,000 

 years. During that time no less than 247 feet would be removed 

 off the general level of the country. 



From the close of the glacial epoch of the Miocene period to 

 the present day, 120 feet of rock must have been removed from 

 the surface of the land and carried down in the form of sedi- 

 ment into the sea. And since the glacial epoch of the Eocene 

 period, no less than 410 feet must have been removed. We need 

 not, therefore, wonder that so few traces of the ice of those periods 

 remain, llemove 410 feet of rock off the surface of the present 

 existing continents, and where should we then find our "roches 

 moutonnees/" striated rocks, boulder-clay, or in fact any evidence 

 whatever on the land that there had been a glacial epoch during 

 the Posttertiary period ? 



XLVIII. Note on the Solvibility of Equations by means of Radi- 

 cals. By Professor Cayley, F.R.S* 

 IN regard to the theorem that the general quintic equation of 

 the 7ith order is not solvible by radicals, I believe that the 

 proofs which have been given depend, or at any rate that a proof 

 may be given that shall depend, on the following two lemmas : — 



I. A one-valued (or symmetrical) function of n letters is a per- 

 fect £th power, only when the Ath root is a one-valued function 

 of the n letters. 



There is an exception in the case k = 2, whatever be the value 

 of n : viz. the product of the squares of the differences is a one- 

 valued function, a perfect square ; but its square root, or the 

 product of the simple differences, is a two-valued function. It 

 is in virtue of this exception that a quadric equation is solvible 

 by radicals; we have the one-valued function (x l — a? 2 ) 2 , the 

 square of a two-valued function x l — x q , and thence the two roots 

 are each expressible in the form 



i {*! + ** + 4/(a?,-a? 2 )8}. 



II. A two-valued function of n letters is a perfect #th power, 

 only when the Ath root is a two- valued function of the n letters. 

 There is an exception in the case # = 3, when w = 3 or 4: viz. 



for 7i=3 we have (on i J r <ox Q -\- o)^r 3 ) 3 (&> an imaginary cube root 

 of unity) a two-valued function, and a perfect cube ; whereas its 

 cube root is the six-valued function x x + ft><r 2 + co~x 3 . And simi- 

 larly for ?? = 4 we have, for instance, 



j x j^g + a? s a? 4 + ft) (a?,a? 8 + x 2 x 4 ) + a^x^ -f x 2 x 3 } 3 



a two-valued function, and a perfect cube, whereas its cube root 



* Communicated by the Author. 



