5t)0 PHILOSOPHICAL TRANSACTIONS. [ANNO 1/84. 



between this sum anditssuccessive one (z-f-e + l)-' . (z + e-f 2) - 1 . ( z -f e-f 3)-' 

 . . . ( z + e + n — I)" 1 X (*(z + l) 6 ' + (3(z + l) 4 ' -« + &c), which will be 

 - (z + e) — . (z + <? + I) - ' •(*+> + a) - " ■• • (z ■ + « + .» - I)" 1 X 

 ((z+e) X («(z + l) A '+P(z +l) 6 '-'+ &c) - (z + e+?i-l) X (az*' + 

 Pz 6 '- 1 -+- &c.) = (/i' — n + l)»z 4 ' + &c.) ; then make the terms of this differ- 

 ence equal to the correspondent terms of the given quantity az 6 + bz h ~ l -\- &c. 

 and there result K = h, — (h — n + 1) X a = a, and consequently a. = 



r-=4- , &c. 



2. Let the general term be (z + e) - ' . (z + e + i) > . ( z -(- e + 2) - '. . . . 

 ( z + e + n _l)-. X ( z +/)-.. ( z+/+ l)-.(z+/+2)-...(z+/ + 

 m — l)- 1 X (az b + /;z fc -' + cz' J - 2 + &c). Assume the quantity (z + e)-' 

 .(z+e-f- 1)-.. . (z+e+n-2)-' X (z+f)-> . (z+/ + l)-». (z + 



/+ 2)-' . . . (z+/+ to — 2) - -' x (*z h ' + Pz 6 '-' + yz 4 '-* -f &c.) for the 

 sum of the series sought ; and thence deduce the general term, which suppose 

 equal to the given general term, and from equating their corresponding parts 

 easily can be deduced the index h' and co-efficients a, (3, y, &c. and consequently 

 the sum of the series sought. 



3. Let the general term reduced to its lowest dimensions be (z -j- e) v X (z + 

 e + ])". . . (z + e + n- l) T X (rz+f)~t X (rz +/-j-r)-e X (rz +/+ 

 2r)~i. . . (rz +/+(/«- l)r)-6 X (z + ^j-X (z+-+ l)-- X . . (z + 

 g+ I— I)"' X &c. X (az b + bz b -' + cz ,J ~ 2 -f- &c). If it be required to re- 

 duce the term (rz -\-f) ~ ? , &c. to a conformity with the rest, for (rz -\-J~)~ ?, 

 &c. substitute (z -f- ) " 5 X r-?, &c. and it is done. Assume for the integral 

 or sum the quantity s = (z + e) v . (z + e -j- l) T . . (z + e -f- ?i — 2) w X 

 {rz +/)"• • (rz + r +/)-i . . (rz + {m - 2r) +/)-t X(z + g)~ x (z + g 

 + 1)-- . . . (z + g + I— 2)~* X &c. X ("Z v + (3z' y -' + &c.) = s, find its 

 successive sum by writing z + 1 for z in the sum s, and let the quantity result- 

 ing be s' ; then will the general term be s — s', which equate to the given ge- 

 neral term, that is, their correspondent quantities ; and thence may be deduced 

 the index U and co-efficients a, (3, &c. ; and consequently the sum sought. If 

 the series does not terminate, then the sum will be expressed by a series proceed- 

 ing in infinitum, according to the reciprocal dimensions of z. 



From t + f + <r + &c. — 1 independent integrals of the above-mentioned 

 kind can be deduced the integrals of all quantities of the same kind ; that is, 

 where h is any whole affirmative number whatever, and the co-efficients a, b, c, 

 &c. are any how varied. If any factor z + g in the denominator, &c. lias no 

 other z -f- g + / — 1 , which differs from it by a whole number / — 1 ; or the 

 factor rz+yhas no correspondent factor rz-\-f-\-mr, where m is a whole 

 number; then the integral of the above-mentioned series cannot be expressed in 



