AliAE NONNtJLL\E APPLICATIO>'ES ETC. 539 



/jys (■^— 7)(^— J— 1 ) • ■ • (.r— J— w-f-1 )_ ■ r(.r— 1 ) . . , (.r—m-^.^ ) 

 ^ ^ 1.2.3...W ~~ 1.2.3...m 



x(:<:.— 1 ) . . . (x — m-^-2) y 

 1.2.3...(m— 1) ■ T 

 3:(.r— 1)...(a-^w-t-3) j >-Cr-t-'i) 

 1.2.3..(m— 2) ■ 1-2 



jO'-<-'l)---(7-f-w— 1) 

 ~ 1.2.3.../W 



Qualuor formulae (I), (II), (III), (IV) direclim c1en)onslralae 

 sunt, quae primo a Lebesgue posilae fuerunt (1) . Eodem pac- 

 10 I'acillime aliae duodecim furmulae deduci possum, quas 

 eodem loco idem Geometra refert , ex quibus elegantes ap- 

 plicationes, et praecipue demonstrationem iheorematis Jacobi 

 deducit . 



7. Ex demonstratis numero tertio habetur aequalitas 



(«-Hl)(n-t-2) . . . (n-t-'»)=y30-i-1) . . . 0-4-m— 1) 



m 

 -*- -^ •/'(/'-1-1) • • • 03-4-/n— 2)X(«— /J-t-l) 



mim — 1 ) , 

 -H \ ^ ./7(^-Hl)...(;)H-m— 3)x(n— y5-t.1)(«— ;7H-2) 



4-(72_p_j-1 )(„_p_t-2) . . . (n—p^m) , 

 Haec formula, posito p-=in, scribi poterit, uti sequitur^ 

 (fl) 1.2.3... («4-to)=7251 .2.3 .. . (H-4-w_i;-t-mx1 .2.3 . . {ti^m—2) 



-t-TOCn— ■')Xl.2.3...()7-t-TO— 3)h- 



^_m(ni—1).. .2.1x1 .2.3 . .. {n—r\. 



Ast e theoria Integralium dejinitorum generatim habe- 

 tur (2) 



(1) Journal dc Matliemaliquds pures el aj)pliquecs par Liouville T. 

 yi. pag. 19. 



(2) Tractatus de Jntegralibus defmitis Celebris Piolae obser>elur 



