510 Prof. Sclionbein on the Polarization of Oocygen. 



74. M, Wantzel's passage from 



</>(^2>^l> • • )=«0(^u ^2» • • ) • • • • (1) 

 to 



(^{oe^yX^. .)=a.(l){x^x^,. .) .... (2) 

 tacitly assumes the wliole question, viz. that the surd is a qua- 

 dratic. The only legitimate inference from (1) is 



d)(«i, a?2, . . ) = «"'^(^2»*u • • )^ ... (3) 

 where «~' is the inverse of a; and if the radical were an n-\c 

 surd, we should have 



</>(^i, x^, . • )=a''-'0(^2J «i, . . )• • • • (4) 

 The assertion that n = 3 must, if admitted, rest upon the 

 ground* that the alternate function or function of differences is 

 the only rational and unsymmetric function which has a prime 

 power (namely its square) symmetric. It may be questionable, 

 however, whether any test could exclude such an expression as 



/(.r,) + ifix,) + i^fjx,) + i^fjx,) + ij{x,) 

 J'{x,) +f{x,) +f{x,) +f{Xs) +/'K) ' 

 in which f'{Xm) is the function (.r— .r„,)-'/(.r), /(.r) = being 

 the given quintic. This expression is affected in value by inter- 

 changes of the x's, although always equal to some fifth root of 

 unity : and there are other indications that, if a finite solution 

 of the general quintic exist, the first radical in the order of cal- 

 culation may be a quintic or other surd. 



4 Pump Court, Temple, London, E.G., • 

 November 9, 1859. 



LXXVI. On the Polarization of Oxygen. 

 By Professor ScHONBEiNf. 



My dear Faraday, 



DURING the summer gone by I have been rather active in 

 my laboratory, and trust my doings will not have been 

 quite useless. I have continued to work upon what I have called 

 " the chemical polarization of neutral oxygen," of which subject 

 I communicated to you something in ray last letter ; and from it 

 you will recollect that during the slow combustion of phosphorus 

 and ether, as well as the electrolysis of water, both kinds of 

 active oxygen (0 and 0) make their appearance, the former in 

 the shape of HO + 0. Having these many years considered the 



* We are in fact remitted to the argument of Abel, or rather to that 

 argument as propounded by Sir W. Rowan Hamilton, who has pointed to 

 a defect in Abel's proof that every radical is a rational function of the roots. 



t Communicated by Professor Faraday. 



