MATHEMATICS, ETC. 213 
in @ a circular substitution cf five letters, replacing x1, wg, £3, 4, Ts, by 
Bg, 3. %,, £5, ©. The function F will not change, and we shall have 
GUE. ha ht Sonat.) == & P'(Lys Va Ce Bas eee e a) 
and, repeating the same substitution, 
Gi Lae ii a es aa at CR cael 0/8) 
with three other equations similarly formed. Multiplying these together, we have 
@ — 1, and this requires a = 1, for a is a third root of unity. Thus the function 
# is invariable for circular permutations of 5 letters,” and, consequently, also 
‘of 3. (Prop. 1.) 
“Thus, all the radicals involved in the root of a general equation of a higher 
degree than the fourth, must be equal to rational functions of the roots, which.re- 
main invariable for circular permutations of three roots. Substituting these 
functions in the expression for 2,, we arrive at an equality of the form 
%, = (@), #,, 43,24, 25, ....) which ought to be an identity; but this is im- 
possible, for the right-hand member remains invariable when we replace 
@y,%_,03 DY Xy,% 3, €,, while the left-hand member evidently changes. It is 
then impossible to resolve by radicals a general equation of the fifth or any 
higher degree.” 
“The preceding demonstration shows at the same time that in equations of the 
third and fourth degrees, the first radical in the order of operations ought to be a 
square-root, and the second a cube-root. These circumstances, in fact, present 
themselves in the known formulas for these equations.” J. B.C. 
THE NEW PLANET. 
M. Le Verrier enjoys the happy peculiarity that his brilliant theoretical dis- 
eoveries are verified at once, and with the most complete and unexpected facility- 
His audacious announcement of the place of a planet beyond Uranus, led to the 
discovery of Neptune on the very evening of ils reception at Berlin, and now his 
still more wonderful announcement of a planet interior to Mereury turns out to 
have been capable of verification before it was made. The following extract from 
Galignani, in default of more detailed accounts, will give some idea of this most 
brilliant achievement, which has at length shot Le Verrier far in advance of his 
rival Mr. Adams, who divided with him the honor of Neptune’s discovery. 
“Our readers must recollect M. Le Verrier’s surprising communication to the 
Academy of Sciences on the 12th of September last, in which he announced a 
-eertain error in the secular motion of the perihelion of Mercury, which could not 
be otherwise explained, than by supposing another planet to exist between 
Mercury and the Sun. Itwould now seem that M. Le Verrier, to whom the world 
owes the unprecedented prediction of the existence of the planet Neptune, has had 
the no less unexampled good-fortune, richly due io his scientific attainments and 
unceasing energy, of seeing his second prediction also verified. The intra-Mereu- 
rial planet has been found. Such is the astounding intelligence announced to the 
Academy by M. Le Verrier himself, and, not only has it been found, but it was so 
several months before M. Le Verrier discovered its existence by calculation ; and 
