4.<y OBSERVATIONFS DE SlNGVLARI 



/ 



cor:c;-l-coi:(4s+ i')+cof.(5;+ 24?)+coC(5;+3v)...+coC(«-f w); 



cor.(s-f-i«^;)rin.^^a=i.'v 

 eiui lumma erit — — ^ — . 



Deinonftratio. 



Variae qiiidem demonftrationes huius Theore- 

 inatis adferri poflent confimiles iis , quas pro Theo- 

 remate primo iu medium adduxi , breuitatis tamen 

 gratia , rimpliciflimam tantum proponere licebit. Po- 

 fita igitur fumma quaefita S, fi progrelfio nollra 

 multiplicetur per 2 fin. \ Vy fiet 



2 S. fin. i-y— 2 cof..s fin.i 17+2 cof. (2-1- v)Cin.'^v+2 coUz+zv) fin. l-y. . * 



-4- 2 cof. (z-{-n v) fin. ' v 

 ir-fin.(5f— ;'z;)-{-fin.(5r-f-ii;)-f-fin.(5rH-|'y) . . . 



- fui.{z+^,v)-fm,{z-\-lv). .. +^^.(.^+(«-1-^)1;), 

 ideoque 

 -fin.(s-!iO+fin.M(«+;Vl cnr.f--4- Jw-u^fin.lCw-l- i)^ 



s- 



afin.ji; fin^v • 



Si in f()rmula allata tam numerator quam denomi-> 

 nator muhiplicentur per fin. 5 1; vel cof. ^ 1; fequcn- 

 tes prodibunt cxprefllones : 



c — c oC z — co/. (g — 1)) -I- ci^r. {z -t-nv) — cof. (r ->- ( n -f-i)v) 



2(1. — COf. VI 



C — — ''m. 'z — V) — fin. z -f - f'". (z -+- n v) -t- (iv. 'z ■^l" -^ r' v) 



j /m. V 



quarum poftericr in fequentcm transmutari poterjt 



c fin.v( — !in.(z — i)i— (w . r- -^ (i".'z -^7iv )-f-lin. '~-t- h -+- Qf) ) 



8. Pro • 



