Infinite Series . 83 
6, If in the denominator are deficient forne intermediate 
fucceffive fadors, multiply both the numerator and 1 denomina- 
tor by thofe deficient factors, and they are fupplied : for exam- 
ple, let A x A"' x A 77777 x &c. be fadors of the denominator, 
in which are deficient the fadors A 7 , A 77 , A 7777 , &c. multiply 
both numerator and denominator by the content A 7 x A 77 X 
A 7777 x &c. and they are reftored. 
7. If in the denominator is contained the content 
A ! xA /l x A 7// . . A 7 ” /w — H, in which A is the leaft of the in- 
dices /, / 7 , / 7/ , &c. : aflume A x x A /X xA 7/X . . . A T * x *x a = H, 
and in the fame manner reduce the fadors in a ; then aflume 
for the denominator A x x A /X x A 7/X . . . A 7k ~ iX x & c. : for exam- 
ple, let the contents be A 3 x A 75 x A 772 x ///S X &c„ = H, then is 
2 the leaft index, and confequently the content reduced as be- 
fore taught will be A 2 . A 72 . A 772 . A 7772 x * = A 2 . A 72 . A 772 . 
A 77/2 xA . A 7 . A 777 x A 72 . A 7772 ; but between the factors A 7 
and A 777 is deficient the fador A 77 ; and between the fadors 
A 72 and A 7772 is deficient the fador A 772 ; multiply the numera- 
tor and denominator of the given fradion by the deficient 
fadors A 77 x A 772 ; aflume for the denominator A 2 • A 72 • A //z < 
X A . A 7 . A 77 x A 72 . A 772 x &c. = A 3 . A 75 . A 775 x &c. &c. 
8 , If the greateft index of the content H is contained in one 
fador only, then the fum of the feries cannot be exprefled in 
finite terms of the quantity 2. 
9. The fame may be applied to the contents of the feveral fuc- 
ceffive values of the quantities B, C, &c. in the denominator : 
for example, let the general term be ^ g 0 ' TI7^ ; mi1 ^” 
ply it into z — 1 to compleat the deficient term, and it refults 
M 2 r 
