REVERSION OF SERIES. 



to defeat, deitroy, or furrender them to him that hath the 

 reverfion ; therefore, when fuch an cftate unites with the 

 reverlion in fee, the law confiders it in the light of a vir- 

 tual furrender of the inferior eftate. But m an eftate-tail 

 the cafe is otherwife : the tenant for a long time had no 

 power at all over it, fo as to bar or to deitroy it ; and now 

 ran only do it by certain fpecial modes, by a tine, a re- 

 covery, and the like: it would, therefore, have been 

 ftranp-ely improvident, to have permitted the tenant in tail, 

 by purchafing the reverfion in fee, to merge his particular 

 eftate, and defeat the inheritance of his iffue : and hence 

 it has become a maxim, that a tenancy in tail, which can- 

 not be furrendered, cannot alfo be merged in the fee. 

 Blackft. Coram, b. ii. ...... 



Reversion of Series, in Algebra, is the method ot finding 

 the value of an unknown quantity, whofe powers enter the 

 terms of a finite or infinite feries, by means of another 

 feries, in which it does not enter. Thus, if we have 

 V = ax + bx + f.v' + dx' + &c. or 

 y = ax" + l>* m+f + c* m+ " + d * m + lP + &c - 

 or if we have 



ay + fiy* + yjf 1 + h* + &c - = 

 ax + bx z + cx s + dx* + &C. 



and we can find in thefe, and other fimilar cafes, 



x= Ay + By' + Cy> + By* + &c. 

 the original feries is faid to be reverted. The reverfion of 

 feries was firft propofed by Newton, in a letter to Mr. 

 Oldenbourgh, at that time fecretary to the Royal Society, 

 with diredions to have it communicated to Leibnitz, and 



in which the author gave one of the earlicft proofs of his 

 great analytical powers. It was afterwards published in 

 his " Analyfis per Equationes Numcro tcrminorum In- 

 finitas," and has fince engaged the attention of many of 

 the moll profound analylls ; and accordingly different 

 methods have been fuggetted for this purpofe ; but that of 

 M. Arbogall, in his " Calcul des Derivations," is the mod 

 complete. We have already, nnder the article Calculus of 

 Derivations, explained, as far as was confident with the 

 plan of this work, the nature of the fymbols, notation, 

 and principles of this dodtrine ; and we may, therefore, 

 under the prefent article, give that author's formula: for 

 reverfion, referring the reader for the firft principles to the 

 article above-mentioned. Still, however, as many of our 

 readers would probably wifti to fee the fame in its plainer 

 Englifti drefs, we propofe, in the firft inftance, to lhew the 

 methods commonly employed for this purpofe by our own 

 algebraitts. This conlifts in afluming a feries of a proper 

 form for the required unknown quantity, and then lubfti- 

 tuting the powers of this feries, inftead of the powers of 

 that quantity, in the propofed feries, and finally equating 

 the co-efficients, whereby the values of the indeterminate 

 er unknown co-efficients, above reprefented by A, B, C, D, 

 &c. will be obtained. 



d* 



faAy + aB 1 

 4- *A« S 



+ aC 

 4- tbA 

 + CA 



y= « 



I 



Confequently we have a A = I, and each of the other 

 co-efficients equal to zero, it being a known property ot 

 two identical funftions, that the co-efficients of the like 

 powers of the indeterminate quantity are equal to each 

 other : and fince, on the firll fide of the above expreffion, 

 y enters only limply, it follows, that all the powers of y on 

 the other fide muft have their co-efficients equal to zero. 

 Whence we have 



«A = i 



nB + bA" =o 



«C + ziAB + CA- = o 



a D + 2*AC + *B' + 3 <A*B + J A* = o 

 Sec. &c. 



and hence, again, we have 



A=i- 

 a 



B = - 



C = 

 D = 

 E = 



ib 1 - 



5 b 1 — S abc + a' d 

 a 1 

 14A 4 - 2iab'c 4- $a % c z + b a 1 bd - a'_* 



+ t« ! + 



-f ex* 4- &c. be 



Let y = ax 4- b: 

 the propofed feries, and 



x = Ay 4- Bjy* 4- Cy' + By" 4- E.y< + &c. 

 the aftumed reverted feries ; then, by fubftituting the feveral 

 powers of this feries, inftead of the powers of * in that 

 propofed, we have 



LBJ,' 



aT> 



zbAC 



AB 1 



3«A«B 



dA' 



4- aE 

 4- 2 ba d 

 4- 2iBC 

 + 3'A'C 

 4- 3<rAB' 

 4- 4^/A'B 

 4- rA 5 



\ y s + Sec. 



And fince a, b, c, d, &c. art known in the original feries, 

 the numeral values of A, B, C, D, &c. in the reverted 

 feries are thus determined. 



With regard to the proper form of the affumed feries, we 

 may obferve, generally, that in order to find the firft term 

 of the reverted feries, the rule is, to fubftitute y" inftead 

 of x in the propofed feries, and to equate the leaft power 

 of y arifing from this fubftitution with unity, which will 

 give the required value of n ; and as for the indices of the 

 other powers they will be the fame multiples of the above 

 value of n, as they are (in the original feries) of unity. 



Let there be propofed, for example, 



= *" 4- bx' 



4- ex' 



4- dx m + >', Sec. 



Bci . 



See. 



Here in order to determine the form of the feries to be 

 aftumed! let s" be wrote for x in the given equation, accord, 

 ing to the ufual method ; and then the exponents, fuppofing 

 * tranfpofed, will be 1, n m, n m + n p, n m + tn 6, n m 

 4- 3 71 /, 8cc. rcfpedtivcly ; of which, the two leaft ( I and 



n m) being made equal to each other, n is found = ^ ; and 

 the differences arc K -^-, - J A &c Whence the ferie. 



to be aftumed is 



..1.. ' ' " ■•*-*> 



x _ e ; + B tv 4- C « V 4- D rTiT + Sec. 



P » (for 



