DETERMINATIO SERIEI NOSTRAE PER AEQUATIONEM DIFFERENTIALEM ETC. 
215 
n. T -P(a+i- r , 6+1-r. 2—r.«) 
_ n (g + 6 — T ) n (—t) n (t—i) y (Q(g — Y -t- n (8 — f + t) i+t-v i 
II(a—1)0(6—l)0(a—1)0(6 — Y ) ~ < OiO(l — f + t)\ ’ 
Hinc formula 86 etiam ita exhiberi potest: 
F{a, fi, — y, 1 — oc) 
0(« + 6 — t)0(— t) X 
— 0 (a — y)0(6 y) 
, 0(a+6—y) 0(—y)n(y— 1) y (0(a+&-|-£)n(6-M:+£) i+i+i 0(a—Y+i)0(6—y-H) i+i—.,) 
“T"n(a-1)0(6—l)0(a-Y)0(6—y) <n(*+<+l)O(Y+A+0 0(i-Y+i)n< * > 
Haecce expressio protinus ostendit, singulas differentias, quae sunt sub signo 2, 
fieri =0, si supponatur y = —k, sed quum hic simul fiat II(y — 1) quan 
titas infinite magna, productum finitum evadere posse patet. Cuius valorem ut 
per quantitates finitas exprimamus, statuamus primo y + k = u>, unde fit 
n (y — 1 )-y-(y-H)(y + 2 ) (y-M — l)u> = 11(0 
sive 
Hir— l) = — 
Rei summa vertitur itaque in eo, ut videamus, quid fiat 
j_ In(a — Y + g-f m)0(6— Y~H + m) l+i-Y+tu OQ* —Y~H) n ( g —Y + *) \+t- T ) 
tu ( 0 {t — Y + 1 + tu) 0 (i + tu) 0(£ — Y + iJOi ' 
si «) in infinitum decrescat. Per principia nota autem hinc resultat 
_ 
dY 
si brevitatis caussa statuimus 
n ( a — Y + <) n ( g —t + 0 r i+i—y TT 
n {t—Y+i) 0(i—k—y) 
solamque y tamquam variabilem spectamus. Sed hinc fit 
^ = —^(a —y + i) ——y + i) + W(i —y + l) + W(i—£ —?) — lo S«* 
Hinc colligitur pro y = — k