21 1 OpUSCULA. 
Ch.(^* — f^f & opportune transferas ternninos, orietur pri- 
mum theorema ; fi vero pro CJT^'" ' fcribas 
C /; . 9'""' . r"- ■A- Sh .(p^ orietur fecundum theorema . 
Simili modo reliqua duo theoremata demonftrabis . Nara 
D C c.(^"' Sc.cp'* = m S c .c^" Cc d Cc.^p-hnCc.c^ . 
Sc.(^^ dc^. Pone JCc.(|D = — -^^ 1, Ji" c . (|3 =: -^— , 
ut oriatur 
C c .(o"" S c . q)" w JTT^""*" ' Cg.(p'""' V<|^-f-« Cg . 
Sc.Q^"'" dc^. Si proJTT^'*"^' fubftituas S c .(^""^ r" - S c .0^ , 
fa^tis congruis fubftitutionibus tertium theor ema fefe of feret ; 
-j- s , m — X n 
fi vero pro C c . (p fcnbas Cc.9 . — i^c.qp ,quar- 
tum theorema apparebit . 
Si exiftcnte ?2 numero integro vel pofitivo , vel negati- 
vo , fit m nuraerus integer affirmativus , in quantitatibus by- 
perbolicis utere theoremate primo , in circularibus tertio . 
Namque fi m fit irapar, & w -4- i par , formula S^^"^^. 
Cq) d(s} dependet S Co dp'y ha:c di S 
- w — 3 
C9 i/^; atque ita deinceps , donec C(^ exponens = o : 
ergo propofita integrado dependebit ab integratione S (o" 
d(^i qu^ in fuperioribus iitteris tradita eft . 
Si vero elfet m par , & m -4- i impar, raethodus produ«; 
cenda efl: , donec deveniamus ad terrainum , in quo C^ ha- 
beat exponentem r= i • ergo propofita formula ab hac depen- 
debit J.tp"^^ C .(|) . ^9 , fed Cq5 . ^/c|) = ri<r<^ ; ergo ultima 
formula fjet rSa^" dS^-, quae femper integrabilis eft , ck* 
cepto cafu , in quo « — o , quia in hoc dependet a loga° 
lithmis . 
Si exiftente m niimero integro vel pofitivo, vel negati- 
vo , lit /2 integer & affirmativus , adhibe pro quantitanbus 
hyperboiicis theorema fecundum , quartum pro circularibus. 
Si n fit impar 5 & -|- i par , demonl\rabis prcpofitam de- 
pen. 
