378 Notices respecting New Books. 



tlie series of coefficients 1, 1, 2, 5, &c. agreeing with the vakies 

 ah-eady found. The expression for the general term is at once 



seen 



to be 



1.3.5...2/n-3^,„_, 

 '^'"= ^727377 



o> 



which is a remarkably simple form. 



2 Stone Buildings, W.C., 

 June y, 1859. 



LIX. Notices respecting New Books. 



An Essay on the Theory of Equations. By G. B. Jerrakd. 

 London : Taylor and Francis . 



THE main object of this pamphlet is to show that the jreraera/ equa- 

 tion of the fifth degree admits of solution. Has Mr. Jerrard 

 succeeded in establishing this proposition ? We shall not take upon 

 ourselves to affirm it. Lagrange, Vandermonde, Euler, Galois, and 

 many other distinguished mathematicians who devoted their powerful 

 intellects to the question, failed in obtaining a direct answer — although 

 the investigations into Mhich they were thus led have made us 

 acquainted with many propositions of great heauty and generality, 

 and have considerably extended the domains of algebra. 



But Mr. Jerrard's result is in direct opposition to a proposition 

 given by Abel, the proof of which, afterwards simplified by Wantzel, 

 has been received by eminent analysts. 



Now it is impossible for both conclusions to be correct ; and 

 having looked very carefully into Wantzel's proof as given by Serret 

 {Algebre Sup&ieure), we acknowledge we can find no flaw in it. 

 Mr. Jerrard's solution runs to such a length, and is besides so intri- 

 cate from the constant introduction of new symbols, that we fairly 

 confess we had not courage to go through the whole of it. Mr. 

 Jerrard is a veteran in this subject, and has done good service. To 

 him we owe a remarkable proposition, which enables us to trans- 

 form any equation into another which shall want the second, third, 

 and fourth terms, or else the second, third, and fifth, by the solution 

 of a cubic equation in the former case, and a biquadratic in the latter. 

 Any investigation of his must therefore not be rejected hastily, even 

 when in contradiction to others. In the present case, however, 

 instead of limiting his conclusion to the statement, " whence I infer 

 the possibility of solving any proposed equation of the fifth degree," 

 it would certainly be worth his trouble to obtain the solution and 

 apply it to one of his own simplified trinomials. 



