276 On the Resolution of Equations. 



They are m coiifornuty to the ideas communicated by the learned 

 Spallanzani inhls httres a VaUisnieri siir Vorigine dcsfontaines. 



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 IGOl, by Bernardini Ramazzini, in his description 

 des font nines 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. 



_ d 



Art. X. — RemarJcs on the Resolution of Eqnatiojis of the fourth 



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



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



The equation y*-f-a3/^+%-f-c=^0 may he resolved thus: assume 



the lunction — -^— — — - — ; p^zy and deter- 



mine S^, S^, and 8^3, so that (B) may be a factor of (A), indepen- 

 dent of x^ y, p, and q^ and we have then 



x'''-{y^-Apy^+4qy+2p^)x^-^{p*-4qyp^+4q^p-{-2 q'^y^)x*-q'^{C) 



x^+yx^^+px+q (fi) 



.(C) ny^x^ 

 put the ratio t^v = ■ , 3 and make y equal to nothing, and we have 



^la —^p^x^ + {p^ -i-4q-p)x^ -q^ ny^x^ 



t^2 



x"^ -\-fx-\-q y'x 



and by composition, 



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



j?3+3/'x^+pj?+y ~ y'x^' (F)' 



but (F) is a factor of a:^^ „^^/4_4py2^4jy/^2p2)j;2 + (p* - 

 4gry'pa +43^^7+292^^2)^4 -gS (E'), and consequently of {E)-(EO 

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

 make x'^-\'y^x- ^px-^-q^^^^ we have also (/* — 4py^ + 45y+wy') 

 x^+45yp*— 2g^y'^=0j but the arbitrary n, allows one h3^othesis, 

 we therefore make y'^ — 42?y^2^4gy/^j^2/^_0^ and then we have, 



by writing for n/ in (E), x'^^ — (y^'^ — 4py^^ +'iqy^+2p^)^^ +(p^ 

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

 the given, y^+ay^ 4-5^4.^=0 we have 4p=~a, (1) 



4q=:b, (2) 



