552 Petri Callegari 



rain r"~n '*~'' "~^ "~- "—2 



n— 4 n— 3 n— 3 



.__{[1(1)_Hl(2)]H-[1(1)l]z 



^ { [1 ''i)-f^'.T.i(-i (3)]-+-[i (■'"■+■1 (2;] ) z"" 



_1- ( [1 (1 )-H-7. . -Hi C3)]-f.[1 Ci"h-1P)] ] z" 



quae aequalitas in banc transformari j'Olest 



n — 1 n — 1 n— 2 n— 2 n— 3 n — 3 n — 4 n — 4 



S„=:[[1(i)]s -(-[1li)]= _[1(i?_h1(3i]= — [10)-h1(2;]z h ]z 



ri—2 n — 2 11— 3 n — 3 n — 4 h — 4 n— 5 n— 5 



— ([1(')]= -1-[1(')]= _[1(i:h-1(2;]z _[1(1)^1(2)]2 _h...]=o 



Exinde patet aequalitas 



(G) S„ =S„_i . s — S„_2 , 



qua celeberrimus Geometra Sturm usus csl causa applicancU 



suum perelegans theorema ad demonstrandum aequaiionem (A) 



per elementum z expressam omnes radices reales habere j cu- 



jus reiGeomelra Gasclieau optimaiii ei occasionem praebuit. (I) 



1 

 21 . Si binomiumj" -i — • per U„ exprimalur, erit 



XJn =Sn S,i — i ., 



Hinc sequilur 



n — 2 -IN Fj — 2 /■r n — i -1 r n — 4 -,v n—i 



(D) U„=[l oi^—CflT'")] Jl (i^^l {■i)])z"~ ^([l (i^ill (2)Ul (iC..-4-1 (3)]), 



_([i (i)-h'!7.Vi c3;]-(-[i (<)-h"T. -h1 (^)])^ 



_t_([l (< )-i-"7. -Hi (^)]-i-[l ('^ T. H-1 (5^]): 



Qua lege haec series iacedat, evidentlssimuni est. Aequalitas 

 (D) , adhibita formula generali (B), ita scribi poterit 



(1) Vide Diarium Matliematicac Liouvillii t. 7 pag. M6., ac. 132. 



