588 PHILOSOPHICAL TKANSACTIONS. [ANNO 1J84. 



then will A -i-5 — - — be equal to the sum of the first In 4- 1, 3« -f ], 



&c. terms in infinitum. This method, in the preface to the last edition of the 

 Meditationes Algebraicae, is rendered more correct and general. 



4. Let the sum of a series required be expressed by a function of a quantity 

 z, the distance from the first term of the series, then will the general term be the 

 difference between the two successive sums generally expressed. 



5. Let the general term be an algebraical function of z: 1st, let it be 



az m + bz m ~' + cz*"- 2 + &c. , , . . 



-7 ; = t, where m and n are whole num- 



~ + e . z + e + \.z + e+2...Z + e + n — 1 



bers ; and m (if the sum of an infinite series of terms is required) less than n by 



, ,, , az m + bz'—' + &c. 



two or more : then the general term — , ■ — • — ■ — 



o z + e.z+e+l.z+e+2 z + e + n — 1 



1 ~' z + e.z + e+l.z + e+2 ~ 



~z + e.z+e+l ' z + e.z + e+l .z + e+ 2 ' z + e . z + e + 1 . z + e + 2 .z + e + 3 



4- &c. . . . : whence, if 



T z + e.z + e + 1 z+e+n- 1 ' 



z+e+2.z + e + 3.z + e + 4 z + e + n — 1 = z*~* + az" -3 + bz"~> + &c. ; 



z + e+3.z + e+4..z+e + 5 z+e+/i— 1 — z"~ 3 + a';"-" + b'z"~' + &c. ; 



z + e+4.z + e + 5. ..z + e+n — 1= z n ~ 4 + a"?" -5 + b"z'— 6 + &c. 



and so on ; then, if m = n — 2, will y = a, 2 = b — yA, i = c — Sa — yB, 



£ = d — ia" — Sb' — yc, &c. ; the integral in infinitum, or sum of the infinite 



series, will be ^2- + g ., +e -' + .« + 1 + »., + «.., + «+ , „ + « + 3 + &C ' 



The reduction of the general term t into quantities of the before given for- 

 mulas was published in the Meditationes, printed in the year 1774. It was be- 

 fore reduced into formulae of the same kind nearly by Mr. Nichole in the Paris 

 Acts. 



2d. Let the general term be t' = 



uz h + bz h ~' + cz*-* + &c. 



z + e.z + e+l.z + e+2..z + c + n-l xz+f.z+f+l..z+f+m—l xz + g.z+g+ l..z+g + l-l x 8cc.' 

 where A is a whole number less than n + m -J- / + &c. (if it be greater, then 

 the fraction can easily be reduced into a rational quantity az h -" -»-'-*c- -|_ & c . 

 and a fraction of the before-mentioned kind) ; then will t' = 



, a I "' I *" I Sr \ 1 r t _i_ ft' | $'_ , 



( z~+~e "TIT/'" 1 " z~+~g + } ~T ^z + e.z + e+i "*" z+f.z+f+ljr z+g.z+g+i "+" 



&C ') ~*~ ^z + e.z + e+l.z + e + 2 "*~ z+f.z+f+l.z+f+2 + z + g.z+~g+\.z + g + 2~T~ C ^" 

 x x' x' 



K z + e.z + e+\..z + e + n-l n z+ f. z+ f + i..z+J + m- \^ z+g.z + g+ ] ..z+g + l-l^^'* ' 

 whence its integral in infinitum, that is, the sum of the infinite series can be 



found when a. = O, *' = O, a" = o, &c. ; and consequently h not greater than 

 n + m -J- / + &c. — 2 ; otherwise not. If h is not greater than n +■ m -(- / -J- 

 &c. — '2, then will «, +■ «' + <*" + &c. = O, for else the sum would be infinite. 

 Let the number of quantities (e,f, g, &c.) be r, then from r independent in- 

 tegrals of a series, whose term is t'; or from (r — 1) independent sums of in- 



