(for it is evident, by infpeftion, that the co-efficient (A) 

 of the firft term muft here be an unit). 



This feries being therefore raifed to the feveral powers of 

 x, in the given equation, and the co-efficients of the homo- 

 logous terms in the new equation compared together, it will 

 be found that, 



REVERSION OF SERIES. 



Whence we have the following values of A, B, C. D, &c. 

 viz. 



A=SL 

 a 



• - I A' 



B = 

 C = 



b_ 

 m 



(i + 



B 



C = 



2 b A B - tA 1 



+ 2 p) b z — 2 : 



D 



(2 nr + 9 m p + g p' 1 + 3 m -f- 6p + I) I' 



D= - 



(1 + m + 3 p) be 



6m' 



+ 



a 



- z b K C - 2 b A T> — $ c AW 

 _ 4^A'B-«A i 



3<A'C- 



&c. 



whence the values of A, B, C, D, &c. become determined 

 as before. 



From the general value of x, found above, innumerable In all the above cafes, the feveral co-efficients a, b, c, d 



theorems, for reverfing particular forms of feriefes, may be &c. are conceived to be totally independent of each other 



deduced. and when this is the cafe, that is, when no fpecific law ob- 



Thus, if .v -+- b x 1 + e .r 1 + dx* Sec. = z ; then (m be- tains between them, it is obvious, that we can proceed no far- 

 ing = 1, andp = l) #is = « — b z + {2 b b — c) z % — ther in the practical folution than we have derived terms in 

 (5 t> % — 5 b <•" + d) s * &c - tIle theoretical inveftigation ; but if, as moft commonly hap- 



And, if x + b x' + c x' + d z 7 + &c. — z ; (m being pens, the feries we are defirous of reverting arife from the ex- 



= 1, and p = 2) x= z — b z' + (3 b b — e) z^ — (12 b' panfion of any function, fo that an uniform law is obferved 



— 8 c b + d)z 1 &c. between its feveral co-efficients ; a fimilar law may frequently 

 Alfo,if. t 4 + bx \ + cx \+ d.J, &c. = *; then (m be- * e dif ™vered in the reverted ferie, though this generally 



ing = I and J =, j \x = a' - 2 \* + (7 bb - 2 b) z* de P Md « » ther . "P°« an mdudion than from any peculiar 



- ( <o % - 18 be + 2 d) .» &c. &c. ***. T wh ? chth * r «erted co-efficients anfe, which is in- 

 It may be obferved, that in all thefe forms of feriefes, the ^ f* &** imperfeaion of tins method of reverfion. 



firft term is without a co-efficient (which renders the conclu- Let US take ' f ° r exam P le > the f e™s 



fion much more fimple). Therefore, when the feries to be 

 reverted has a co-efficient in its firit term, the whole equation 

 muft be firft divided thereby. Thus, if the equation was 

 3 x — 6 x 1 -f 8 « 5 — 13 * 4 &c. — y ; by dividing the 



whole by 3 it will become x 



2X Z + 



'3* 



&c. 



z + 22' + 



2y' 16 v ! 

 9 81 



3 3 



= 4_y ; where, putting z =tJ, we have, by For 1. x 



— a 3 &c. = ^~ 

 3 3 



When there are two feries, confiding of like powers of .v 

 and y, as 



a x + b x 2 4- c x* + &C. = ay -f /S y 1 -f y y'< + &C. 



aflume, as in the preceding cafes, 



x=A^ + B/+ C / + D y + &c. 



and let his, and its powers, be fubftituted for x, and the 

 powers of x, an'' we lhall have 



a. y 4- f3_y* + y y 2 -f $y" + SiC. = 



aAy + a~B\ , + aC ~) + a D ~] 



bA'P +2bABW+ibAC\ 



+ 'A' J + b B> }>/ + &c. 

 + 3 ' A*B 

 + dA> J 



in which, inftead of equating all our co-efficients to zero 

 they muft now be equated to a, 0, y, J, t , Sec. ; that is 



a A = a 



a B + b A 2 = $ 



«C +2bAB+cA i = y 



a D + 2£AC+JB I + 3<rA*B + d A* = I 

 &c. &c. 



x' x' *' x> 



H H 4 H &c. = z, 



2 3 4 5 

 to find the value of x in terms of z. 



This agrees with our firft form ; where a = 1, b = h, 

 c = f , */ = ^, &c. and we have 

 1 

 a 



2 



+ (2^-^) = 



I 

 6 

 1 



1 

 2^3 



- (5 & - 5 a b c + crd) = — = 



24 



+ (14 b* - 21 a b z c + 3 a 1 c* + 6 a* *</- 

 I 



-7',) = 



I 

 120 



3 -4-5 



And hence, inferring the fame law to have place through- 

 out, we have 



x = a — 

 &c. 



** + 



_ 2 4 + 



4 



3 -4-5 



We muft not, however, look for the like uniformity of 

 refult in all cafes. As an example of the contrary, let there 

 be propofed the feries 



2 2 



- X' 



3 2.3.4 



** + 



= 7 y + -f + -f + -y + 



2 3 4 5 



2 -3-4-5 

 ^-y' + &c. 



to find a- in terms of y. 



Sic. 



Here 



