61 



REVERSION OF SERIES. 



REVERSION. 





don (to keep the common term) will be met with in TAYLOR'S 

 THEOREM and SERIES ; the present article U meant purely for reference 

 npon the most usual case of the problem, which is not sufficiently 

 developed in elementary works; that is, enough of the result for 

 reference is not put down. 



The problem is as follows : Given y = oj + ix 5 + ex 1 +ex* + 

 fjf ...; required x in a series of the form Ay - By' + cy* ty 4 -<- 

 .... It will be proper first to put down the coefficients in connection 

 with the exponents to which they belong, as follows : 



123456789 10 11 

 a h i- e f g k t I w 



ABCRFKHKL M X 



H will be convenient, instead of writing the resulting series 



- By- + ..... , to let it be AO-'y Ba~ 3 y" + co-'y 1 Eo-~y' + 



- - - 



* 



ro-V - 



s-"y 



A = 1 



- so-" y 

 We then hare 



f - 



E = 54* - Sabe 4- tfe 



r = 144 - 21o4e + 8a(24 + r) - iff 



o 424 - 84a4e 4- 28 (AV + A>) - 7a(A/ + ) + ay 



11 = 1S2M 580o4'e + 0a l (24>e + 34V) - 12a J (34= / + 



: ^7o4V 4- 495a' (4'e -r 24V) 165o> (4/-f 



V) + 45a (*-./ + 2A</ + 4e + A) - o (4* + ey + ef) + aft 



i 14304 - 5005oAc + lOOla 1 (2AV + 5AV) - 715O 1 (A 4 / + 44Ve + 



24V) + 55o 4 (4*V + l - b ' f f + MV + 124<:i!e + ') ~ 55a * C* 1 * + 

 2Arj + 24gr + />) -r I* 1 (24t -r 2A + 2q + /*) - a'' 



* = 42A - ]448a4V + SOOSo' (4 + S4V) - 1001a (34/ + 

 15A< + 104V) * lOOlo'tf'j + 44V + 2AV + 4 A + 4r) 

 28oa (A" + Vft; + S4V/+ ***/ + V>et> + c>e) + 22o (M'X- + 

 AA + *>y+3cV-f3A/5 + */+,) _ Ho' (M + e* + * + 

 /o) + *s. 



= 167M4" - 75582oMe + 1591ii : (-J',v *- Ti'c 5 ) - 12376o>(///+ 

 + 5AV) + 2184 (24y 10/,<-/ + 64'e + 20*>c' < + 

 84V) - 27J (5Wk + 204<y + 204" if 

 204A + eO + 182* (34* + 8AV& -I- Wtg 

 124/-(- 2c"/+ 24e S* 5 * 8 ) - 78o' (4 s ! + 

 + tify -t- 2!/ + </ + /) + 



34-/- 



We have given these coefficienU to an extent which many will think 

 useless, and in fact it will not often be necessary to employ all that are 

 here given. But we have two objecU in view : first, to enable those 

 who want these coefficienU to refer to them ; secondly, to point out 

 the great advantage of some methods which are never given in ele- 

 mentary works, ana are not so much known and practised as methods 

 of such utility and power should be. 



The usual way of obtaining thise results is to take the series 



f m A*T~' f Bar" 4 ** + , and in it to substitute the value of y, 



namely, e-r + 4x" -r This would give 



x - 5 <r + I* + ...)- 5 (ox + Ax' ...)'+..., 



the two sides of which equation most be identical ; giving 

 4 1 



24 



and so nn ; but this process would become intolerably tedious and 

 liable to error after a few steps ; that which we have followed in 

 forming the preceding coefficients [TAYLOR'S THEOREM] would have 

 enabled' as with comparatively little difficulty and small risk of error 

 to double their number. It also gives the law of the coefficienU, 

 which u as follows : 



1. What sort of terms enter into M.the coefficient of y*? Write 

 down every way hi which 2(10 - 1), or 18, can be made up of 10 I, 

 or 9, numbers, and, taking the letters belonging to these numbers from 

 the table, we have the literal part* of the different terms of the 

 coefficients. Thus 



18 is made up of the nine numbers, 1,1,1,1. 2, 2,2, 2, , the letters 

 of which are a,n,n,n. '., //, h. A, g : accordingly, o 4 A 4 g is the literal 

 pert of one of the terms of M. And similarly for every other com' 

 bination of nine nnml>em which make IS. 



2. What is the coefficient of any given term r Ray that y* is the 

 power to which the term belongs, and that a'l^r* .... is the literal 

 pert of it. The coefficient required is as follows : 



( + 1) ( + 2) (2-o-2> 



(1.2.8....0) (171717.. .T) (1.2. 8... .).... 



Thus, to verify tho numerical coefficient of afife- in x, the coefficient 

 of y", we must calculate (n = 11, o = 6, 2n o 2 = 14), 



12 . 13 . 14 

 (1 2) (1 2)' wulc ' 1 fa 8 18 . 7j or 516, 



and 1S2 x 3, the coefficient iu the table, is also 546. 



2. The sign of auy term is positive or negative according as the 

 power of a. which it contains is even or odd. 



We may thus verify any one term, and the coefficients may be 

 sufficiently verified, as to typographical correctness, by remembering, 

 that if a = A = c = 4c. = 1, we should have A = B = c = &c. = 1 ; 



for y = x + x- 2 + JC* + . . . gives j- = y y- + y 1 The result 



f the use of the preceding table, distinguishing the positive from the 

 negative parts, is 



x = y - y> + (2 - 1) ./> - (6 -5)y + (23'- 22) / - (99 - 98) 

 + (452 - 451) y* - (2140 - 2139) y 8 + (10397 - 10396) " - 

 (51525 - 51524) y 10 + (259430- 259429) y. 



The preceding is a particular case of the following general problem, 

 which frequently occurs, and is very complicated in its details. Given 

 y m = ox" + AJC"+ I + ex"+ s -r M" +> + /r"+ 4 + ....; required a series 

 for . in powers of y. Let it be assumed that 



f F- 



a* 



n being 1 : M. Then A = 1, B = n4 : 



3n + l 



= "~2~~ 



4 f 1 In < 2 4* + 1 



E = a 3 g- A> - 5 i 



5 + 1 5M + 2 5* +3 

 v = M -r-T- s r- 6* n 



+ ' 



5n + l 

 + *^- a* (2 !* + <*) -no'/ 



Methods of obtaining all these series are given iu TAYLOR'S 

 THEOREM. 



REVERSION. " Reversion of land is a certain estate remaining in 

 the lessor or donor, after the particular estate and possession conveyed 

 to another by lease for life, for years, or gift in tail. And it is called 

 a reversion in respect of the possession separated from it ; so that he 

 that hath the one, bath not the other at the same time, for being in 

 one body together, there cannot be said a reversion, because by the 

 uniting, the one of them is drowned in the other. And so the 

 reversion of land is the land itself when it falleth." (' Termes de la 

 Ley.') Thus if a man seised in fee simple conveys lands to A for life, 

 or in tail, he retains the reversion in fee simple. The distinction 

 between a remainder and a reversion has been explained in REMAINDER. 

 In all cases where the owner of land or the person who has an estate 

 in land, granU pert only of his estate, he has a reversion ; and as the 

 grantee holds of him, there is tenure between them, and the grantor 

 has a seignory by virtue of having a revt-r i n. When a man grants 

 all his estate to another, or grants a particular estate to A, and various 

 remainders over, remainder to K in fee, he has no reversion left, and 

 therefore he has no seignory since the passing of the statute of Quia 

 Emptores. The remainder-men alxo who precede tho remainder-man 

 in fee, do not hold of such remainder-man, but of the lord of the fee 

 of whom the original owner held. The word reversion is often used 

 .inaccurately, and it is sometimes necessary to recur to its strict legal 

 signification. 



Before the passing of the statute De Donis, if a man seised in fee 

 simple granted his lands to a man and the heirs of his body, he had no 

 reversion, for the grantee was considered to have a conditional fee'. 

 But since this statute, an estate to a man and the heirs of his body has 

 always been considered to be a particular estate. 



If a man grants a lease of lands in possession, at common law, he 

 has no reversion until the lessee enters by virtue of his lease, fur the 

 lessee has no estate until he enters ; but if the term of years is created 

 under the Statute of Uses, as by bargain and sale, the lessee has a 

 vested estate by virtue of the statute, without entering on the land, 

 and consequently the lessor has a reversion. It is said that a reversion 

 cannot be [created by deed or other assurance, but arises from con- 

 struction of law. This means that a reversion is not created by the 

 act of the party who conveys part of his estate, but is a legal consc- 

 of his acts. If a man seised in fee simple limits his estate to 

 another for life or in tail, remainder to himsvlf in fee or to his own 

 right heirs, he has not a remainder, but a reversion. Yet by 3 and 1 

 Win. IV. c. 100, the effect of such a limitation is to vest such a 

 P'tnainder in fee in the settlor by pnrchate, and he is not to be con- 

 sidered to be entitled to it as his former estate or part thereof. 



A reversion is a vested estate, which may be granted or conveyed, 

 and charged like an estate in possession; and hi some cases the 

 reversioner in fee may bring an action, as well as the tenant in 

 possession, for an injury to his inheritance. 



