VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 589 



finite serieses, whose term is r' ; that is, where h is not greater than n -f- m + I 

 -f- Sec. — 2 ; can be deduced the sum of all infinite serieses of the before-men- 

 tioned formulae, whose general term is t'. 



If any factors are deficient in the denominator, as suppose the term to be 

 z-|-eXz + e-f-3Xz. + e + rc— 1; multiplying the numerator and denomi- 

 nator by the deficient factors, viz. by z -f e + l.z + e + 2Xz + e + 4.z + 

 e + 5. . « + e + n — 2, and it acquires the preceding formula ; and so in the 

 following examples. 



3d. Let the denominator be (-r + e)« X (x + e + l) T X (r + e + 2) T 



(a. + e + n - l) T X {x + e + ,*)*' X (x + e + ,* + 1)"' X (ar + e + ,* 4- 2)"' 

 X &c. X (a +/> X (x +/+ l)c x (* +/+ 2)(. ... X (* + /+ m - l)« 

 X &c. = d, where ir, ir', g , &c. ; f*, &c. are whole numbers ; and the general 



term is — - - ! = t" ; then, it the dimensions ot z in the nume- 



D 



rator be less than its dimensions in the denominator, will t" = 



(z+ e + (z + ey + (i+ey-- {z + e) : + *' + Z +f + (•+/)» + (l +/) 7 * ' * " 



— — , + &c.) + ( - 1 ^— -. 4- &c.) + &c; and in ge- 



(z+J)*^ J ^ ^z + e.z + e + 1 ~ z+j.z +/+ 1 T / T~ » & 



neral there will be included all terms of the formulae, 



(z + e + iy - (z + e)> {z+ f +i y'- {z + fY 



(z + e)" . (z + e -h 1 )" . . . (z + e + i)> > (: + /)"' . (z +/ + 1 )"'. . . (z +/ + i")"' ' 



_ %^ e ( \ yJ Y Y '~S, + e t + I: Y ' ^,c, &c. where a, b, c, &c. «, «', 



(z + e + ^y . (z + e + f + 1 ) p ■ . . (z + e + p + t)r" ' ' ' ' ' 



&c. (3, (3', &c. y, 0, &c. denote invariable quantities ; and p, p', p", &c. are 

 whole numbers not greater than v, % , ir', &c. respectively ; and i, i', i", &c. are 

 whole numbers not greater than n — 1, m — 1, &c. 



If all the quantities x, a.', a", &c. (3, [3', (3", &c. &c. are = O, the sum of the 

 series can be expressed in finite terms of the quantity z, otherwise not ; and also 

 if h be less than the dimensions of z in the denominator by 2 or more, then 

 will x + (3 + &c. = O, otherwise the sum would be infinite. 



From 7T 4- 7/ + f + &c. — 1 independent sums of infinite serieses of this 

 kind can be deduced the sums of all infinite serieses of the same kind. This 

 method may be extended to infinite series, in which exponentials, as e% are con- 

 tained, which will easily be seen from some subsequent propositions ; but in my 

 opinion the subsequent method of finding the sum of serieses is to be preferred 

 to the preceding one, both for its generality and facility. 



6. 1. Let the general term be (az b + bz h -' + cz b -* + &c. X (z + e)~ ' . 

 (z 4. e + 1)1. (z + e + 2) - ' . . . (z + e + n — 1 ) " ' ; where A is a whole 

 number less than n by 2 or more, when the sum of an infinite series is required. 



Assume for the sum the quantity (z -j- e) -1 . (z + e + 1) _I . (z + e + 2) _I 

 . . . (z + e + n — 2)-' X (*z b ' + (iz h '~ ' + yz h '~ z + &c.) ; find the difference 



