276 Oil the Resolution of Equations. 



They are in conformity to the ideas communicated by the learned 

 Spallanzani in his lettres a Vallisnieri sur Vorigine des fontaines. 



The explanation which has been given of the overflowing of the 

 water from shafts made by boring, is in conformity with that which 

 was published in 1691, by Bernardini Ramazzini, in his description 

 des fontaines jasillisantes de Modine, a work which is now very scarce, 

 and what is still more remarkable, the author, in explaining the theory 

 of these fountains, which were then considered as wonders, proves, 

 that he was as good a natural philosopher, as a geologist, and that he 

 possessed very superior knowledge, for the time in which he wrote. 



Art. X. — Remarlcs on the Resolution of Equations of the fourth 

 degree; by Mr. C. Wilder, of New Orleans. 



(See Vol. XVI. p. 271, of this Journal.) 



The equsxlon y* -{-ay ^ -\- by -\-c=0 may be resolved thus: assume 



, , ,. ^-+S,a.«-}-S3^+S,3 (A) 

 the lunction r,—^ — ; ToV and deter- 



x-^-\-yx'' -}-px-\-q [ti) 



mine S^, Sg, and S , „, so that (B) may be a factor of (A), indepen- 

 dent of X, y, p, and q, and we have then 



aj' '^-(y'^-4py'^+4qy-h2p^)x^-\-{p'^-4qyp^-{-4q^p-\-2q'^y^)x'^-q'^[C) 

 x-^-\-yx"-{-px-{-q (D) 



(C) ny'x^ 

 put the ratio rn\ = ~rT~ ^^^ make y equal to nothing, and we have 



x^'^—2p-x^-\-{p'^-{-4q'^p)x'^—q'^ ny'x^ 

 x^ ■\-px-^q y'x^ 



and by composition, 



x^^ -{2p^- -ny')x^-{-{p'^-\-4q-p)x'^ — q'^ ny'x^ (E). 

 x^-\-y'x^-[-px-\-q ~ y'x^ (F) 



but (F) is a factor oi x^^ —{y'*—4py'^ -\-4qy'-{-2p'^)x^-\-(p* — 

 4qy'p^ -{-4q^pi-2q^y'^)x^ -q*, (E'), and consequently of (E)-(E') 

 or of (y'* —4py'^-{-4qy'-{-ny')x^-\-{4qy'p^ —2q^y'^)x'^ : now if we 

 make x^ -\-y'x^ ■\-px-\-q=^0, we have also [y"^ —4py'^-\-4qy'-\-ny') 

 x^-\-4qyp'^ —2q^y'^=Q; but the arbitrary w, allows one hypothesis, 

 we therefore mdke y"^ —4py'^ -\-4qy'-\-ny'=0, and then we have, 

 by writing for ny' in (E), a?^ ^ — (y'*—4py'^ -\-4qy'-\-2p^ )x^ 4-(p* — 

 4q^p)x'^ —q*, dropping the accent, and comparing this equation and 

 the given, 2/*-f'fi^2/^+%+^==^ we have 4p= — a, (1) 



4q=b, (2) 



