VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 5G|3 



8. 3. If the terms of the series assumed -— + — • — ■ 1- 



e r + e 2r + e 3r + e 



&c. be alternately affirmative and negative ; then by the preceding case find 



az m + 0-m-i + yz m-t + &C- a j, c 



rz + e . rz + r + e . rz + 2r + e + &c. rz + e rz + r + e rz + 2r + e 



Where the terms of the resulting series are alternately affirmative and negative, 



let the two subsequent terms be supposed r: + £ , r; + ,. + e , , . r2 + ( „_ t) r ~ e 



a 6 . , «(z + l) m + 3( z + l) m -' + v(z + 1)"-* + &c. 



= h &c. and {■ — ; , a . . ; = 



rz + e ~ rz + r + e' rz + r + e . rz + 2r + e . . .rz + nr + e 



. 1 _ u &c. of which the one is affirmative and the other ne- 



rz + r + e ' rz + 2r + e 



gative : reduce the resulting series to an affirmative one by subtracting the sub- 

 sequent term from its preceding, and it becomes 



/ rz + nr + e) . («z™ + fl;"" ' + &c.) - (rz + e) . («(z + l) m + Hz + 1)"— + &c.) 

 rz + e . rz + r + e . rz + 2r + e . . . rz + rn + e 



("-'"W-+&C. __ _«_ + _ 6 -Z^_ + & c . In this case, 



— rz + e .rz + r + e ..rz + rn + e rz + e rz + r + e 



since two terms are added into one, the distance from the first term of the series 



will be - , which suppose = iv ; and write 1w for z in the above-mentioned 



, , (« — m)r «.z m + &c. 



term, and there results ^ - e .rz + r + e .. .rz + nr + e 

 = (» - ™)r« x W + fc*_ = . *- .a &(j 



— 2rw + e . 2rw + r + e ... 2rw + nr + e 2rw + e 2rw + r + e 



, . ■ a'w m + h'w m -' + &c. 



the sum of any series, whose general term is 2ra)+ e , 2rw + f +e ... 2< . B) + >tr + - g 

 where m is a whole number less than n by 2 or more, and w the distance from 

 the first term of the series can be found from the sum of the series 



I _ _L_ + _J I 1 — + &C. 



e r + e ' -2r + e 3i ■ + e 



g. Let there be two serieses \ + — — + — ^ + &c. = s andj. + ~- 

 ■ ] i 1 l &c. = s\ whose general terms are respectively-) 



T/+ 2r ^f+ 3r T & « + rz 



aw \ _i_ -J— • then from the sum of these two serieses can be collected the sum 

 of any series, whose general term is 



J «.z m + Sz m - ' + &C. 



fz +e r- + e+r.rz + e+ 2r...rz + (n- 1) /■ + e x rz+f.rz + r+f. ..rz +/+(«i-J)r 



a , b . c_ , * , J__ , 



= rT +~e -T r z + e + r ^ rz + e + 2r y ^ rz + (n - l)r + e Tz+f^ 



b ' i £ J — - r ; where e —f is not a whole 



, , T~ ~f i rz j. gr + f rz + ( m ~ J ) r +/ 



number.' Leta + b + c . . + * = 0, and a' + b' + c' . . . + ?' = 0, then the 

 sum will be a (— — e + ~ + r + e •••+ rz + (n - 2)r + e> + ' Vi"+7+~r + 



7z'+l+ 2r • • • + ST5"- S)' + e ^ + C ( '"Te + 2/' + ~~e~^ + • • • + 

 VOL. XV. 4 Cr 



