
392 | Dr. WariNne on the. ! 
3. Let the general term reduced to its lowett dimentions’be ‘i 
Zoe gay mp ee SHEtu—I ee x PEAS EP st 
———_——— : i a Oy TR ———— 
retfror >... retftm—ir Koto “x 2 oes ve ae 

ope. oe ace x (az? ee a +&c.). If it be re- 
quired to reduce the term rz + vee &c. torr sofort with 
the reft, forrz+f °, & _paleoaeuee Bb i} ‘x4r-e, &c. and it 
is. dene. .Affume for the integral ‘or * fie ‘the quantity 
Saepessbebi .. ebetn—2 x reef \. ree Pap, 
Pet m—art fox She XZ4Sh1 sy Spee ee 
x (ae” + 0x’ + &c.) =S, find its fueceflive fum by writing 
z+1:for sin the fum S, and let the quantity refulting be 
S’; then will the general term be S—S’, which equate to the 
given general term, that is, their correfpondent-quantities ; 
and thence may be deduced the index 4’ and co-éfficients a, 6, 
&c.; and confequently the fum fought. If the feries does not | 
terminate, then the fum will be expreffed by a feries proceeding 
in infinitum, according to the reciprocal dimenfions of 2. 
From 2w+e+o+ &c. ~1 independent integrals of the 
above-mentioned kind can be deduced the integrals of all quan- 
tities of the fame. kind ; that is, where 4 is any whole affirma- 
tive number whatever, and the co-efficients. a, 4, c, &c. are 
any. how varied. 
If any factor .z+g in the denominator, &c. has no ote 
z-+g¢+/—1, which differs from it bya whole number/=13 or 
the factor rz+.f has ne correfpondent factor rz +f-++ mr, where 
mis a whole number; then -the integral of the above-mentioned 
feries cannot be exprefled in finite terms of the quantity 2. In 
like manner, if the dumenfions of z in the numerator are lefs © 
than 

