59 



A~ ^ 



m > 



m 'vTn + n.) ' 



Q — - _^ X(X— I) na . 



m (77i+-n) 1 m -t-2 II) ' 

 £j — _ X(X — I)(X— 2 ) n3 , 



tJi (m -H n) (m + 2 n) (m + 3 n) ' 

 £ -— I X(X — I) (X — 21 (X — 3)>i4 , 



T» (m + n) (m + STi) (mf 3n) tm + 4fi) ' 



etc. etc. 



qui si substituantur in serie pro v assumta , formula integralis 

 proposita 



S zr /x™~^ 3 X (A -H x^)^ 

 seo[uenti modo per seriem semper convergentem erit expressa: 



r j . Xn r_x"__-\ _^ Xn (X— l)n /- x" y 



Q .yWf \ Jt-x")^ j m-i-n ^A-t-x""' m-t-n * m-)-2n ^A-|-3c" 



b . — < ^„^ ^ (X-I)n ^ (X-2)n r x" y i gj-^ 



( m+n * m-)-2n ' m-H3n ^A-f-x" ' * 



quae cum Euleriana supra §. 3. exhibita perfecte congruit, sem- 

 perque manifesto convergit , quicunque valores variabili X et 

 constanti A tribuantur. 



§. 7. Sumamus X ~ — i, m — i, n zz 2 et A ir: i, 

 eritque formula integralis proposita 



S = /j^-. = A. tag. X 

 series vero convergens hoc casu fiet 



S^^l^xx^ 3. 5 '^l-t-arx^ 3 5- T^I + xx'' 



, 2^6_8 c^__y ^_ 2_i.6_8_ig /'__fii_y _^etC. 



3- 5- 7- 9 ^I-f-x:« 3 5. 7- 9- II ^I+^x'^ 



quae -est ea ipsa series, qua Eulerus usus est ad eas series ma- 

 xime coRvergentes inveniendas , quibus ratio peripheriae circuli 

 ad diametrum vero proxime exhiberi potest. In dissertatione 



H 2 su- 



l-hxx 



