5^8 PHILOSOPHICAL TRANSACTIONS. [aNNO 1789* 



Exam. 1 . sn = ns (n — 1) — n . ^^^ s{n — 1) •\- n . ^— . ^^~- s (n — 3) 

 . . . + wsl + s nearly. 



£^GW. 2. s (n + w) = —^ ]~~3 — ;riri X s (n — 1) — 1— i 



XAX^4xs^(«-.) + "-|^XBx'^s(«-3)-"-^XcX^ 



X s (n — 4) + --T~ X D X ^-T"5 X s (n — 5) — &c. nearly, where the letters 

 A, B, c, D, &c. denote the preceding co-efficients, and the converging series is 

 the same as in the preceding example. 



Exam. 3. Let the converging series be of the formula ax -j- hoi? X cx^ + dx'' -f- 

 &c.; then will m — {2n — 2) s (w — 1) — {2n — l) X ^^^s (w — 2) + (2n 



, \ v> 27e — 2 . 2ra — 6 , . . ,^ , v 2ra — 2 2« — 3 ^^ 2/1 - 4 . v 



- 1) X —2— X -Y- s (w - 3) - (2n - 1) . -^— . — — X ~— s(n--4) 



4- &c. nearly, of which the general term is (2n — l) . 



2n — 2 2rt — 3 2« - ^ + 1 ^^ 2» — 2/ . . ^ , ;^ 



These series may be made to begin from any term, which may be easily found 

 by the method of correspondent values, and the subsequent terms from it by the 

 given law; its preceding terms may be deduced from the same law reversed, that 

 is, by putting the numerators of the fractions multiplied into it for the denomi- 

 nators, and the denominators for the numerators. From these different serieses 

 may be formed, by adding 2 or more terms of the given series together for a 

 term of the required series; which method has been applied to converging series 

 in general in the Meditationes. 



13. The method of correspondent values easily affords a resolution of the 

 problems contained in Mr. Brigg's or Sir Isaac Newton's method of differences. 



Exam. 1. Let the quantity be of the formula a -\~ bx -\- cx^ + da^ + &c. . . 

 a?" = 1/j and n + 1 correspondent values of x and z/ be given, viz. p, q, r, s, &c. 

 of .r; sp, sg, sr, ss, &c. of 7/; then will t/ = 



X — a . X — r . X — s . &c. ^ , , , x — p . x — r .x — s , &c. ^ ^ , x — p . x — q .x -- s .he 



^ X— X &p-\ o- X SO H ^ 2 tl^±. 



p — q.p — r.p — s.eic. ' ' q — p .q — r , q — s . oic. ^ ^ — P -^ -^ q -r — s . Sec. 



X sr + &P- The truth of this problem very easily appears by writing p, q, r, s, 

 &c. for X in the given series. 



All the preceding examples may be applied to this case, by writing x for m in 

 the given series; hence the resolutions of several cases of equi-distant ordinates 

 by easy and not inelegant serieses, among which are included the 2 cases com- 

 monly given on this subject. 



14. If a quantity be required, which proceeds according to the dimensions of 

 x'y reduce the above given value of y into a quantity proceeding according to the 

 dimensions of x, and there results y = 



y—q.p — r.p — s.&cc. z=i a.*" q —p.q — r .q— s .S)CC. = ^' ^ a "^ 



