REV 



i i , i 



Here a — I, b = -, c = , a — - — - — -, &c. 



2 2.3 2-3-4 



«= -, £ = -, y = -, i = -» &c. 



2 3 4 5 



whence 



--=1= A 



a 2 



/S-iA 1 



1 1 



B 



= C 



a 24 



y- iiAB -c A' _ 



a ~ 24 



3-6B' - 2^AC-3cA'B -</A_ *_ 138 1 



a 2880 



( -2JBC-2iAD- jtAB'-iic, _ 



D 



1543 



3840 

 '543 . 



= E 



whence the propofed feries is 



1 11 11 , 1581 



2 -^ 24 J T 24-^ 2880^ T 3840-^ 



in which no law of continuation can be difcovercd, either by 

 induction, or otherwife. 



In this refpect the fymbols and notation of Arbogalt Iiave 

 a decided advantage, as by thefe means the law of formation 

 in the reverted feries is exhibited in the cleared poflible point 

 of view. Taking for example our firil feries, under the 

 form 



y = $ x + y x* + 5 x~' + e a- 1 4- &c. 



to find x — b y + cy* 4- dy 1 -f c y* + Sec. 



Here the general refult is exprefled by 



.v=0- , y + -.V.t 





4 L 



. / + &c. 



where the law is clearly exprefled by means of the fymbol D ; 

 and if it be requifite, the co-efficients may be eafily exprefled 

 in terms of 0, 7, S, &c. thus/3-' = £—. 



3-c^- 3 = 7h- D (-3^ 4 ->) 



(-3.-4^-'.-/- 3 ^- 4 -2^) 



2y a -/94 



« D' 



4 = 



ii 



= I ( 4 ^- 5 .* + 4 -^0- 6 -2y,S 

 4 1.2 



1.2.3 



— 5&y*— Sy 3 — P' e 



which will be found to agree with our former refult. 



REV 



We refer the reader who may not be acquainted with the 

 ufe and Ggnifieation of the above fymbols and notation, to our 

 article Derivations. 



In thefe formula; the quantities ,?, y, ,', Sec. although 

 totally independent, are yet apparently connected by means 

 of the fymbol of derivation D ; but the application is gene- 

 rally made to feries which exprefs the evolution of fome func- 

 tion, and the co-efficients of which feries are COnfequently 

 formed after a certain law : for inltance 



I + x + 



+ ■ 



1.2 I.2.3 



is the evolution of c r , and if y be put 



+ &c. 



= 1 + * + 



+ 



1.2 1.2 



then, by the theorem for reverfion, 



+ &c. 



= 0- (y 



z) + i D., 

 ■(>>- 



(-». (y- i) 1 + 

 1)' &c. 



but in the propofed form 

 jS = 1 . D . (3 = 



I)' 

 C ' 



ID 1 



■> C ' 



&c. 



therefore x = (y — i) (y — i)* -\ (y— i) J — &c 



And in the fame way from reverfion, if 



+ &c. 



by reverfion 



x — % — 



+ • 



3-4-5 



&c. 



We refer the reader for more on this fubject to Wood- 

 houfe's " Principles of Analytical Calculation," and Arbo- 

 galt's " Calcul des Derivations ;" and for the principles of 

 the method Hated in the former part of this article, to Simp- 

 fon's " Fluxions," vol. ii. p. 302 ; Maclaurin's Algebra, 

 p. 263. See alfo Newton's " Analyfis per Equationes," 

 aad Bonnycaftle's Algebra, vols. i. and ii. 



REVERSIONS, in the Dotlrine of Annuities, are either con- 

 tingent or abfolute. (For the former fee the article Sur- 

 vivorships.) Of abfolute reversions, the cafes are very 

 few, and the folutions fimple and eafy. An abfolute rever- 

 fion, whether it is to take place after the extinction of a 

 fingle life, or of any number of lives, or after the expiration 

 of a given number of years, mult necellarily be more valu- 

 able, in general, than a contingent reverfion ; which, depend- 

 ing on events altogether uncertain, will be of lefs value in 

 proportion as thofe events are lefs probable. The following 

 problems include the principal cafes of abfolute reverfions, 

 and their folutions being almolt felf-evident, require no ex- 

 planation. 



Problem I. 



To find the value of the reverfion in fee of an annuity 

 after a given number of years. 



Solution. — Deduct the value of an annuity for the given 

 term from the perpetuity ; multiply the remainder into the 

 annuity, and the product will be the value required. 



Example. — Lei the annuity be 15/. the term ly years, 

 and the rate of intcrelt 4/. percent. By Tab. III. (fee 

 Annuities) the value ot an annuity for 15 years ia 

 1 1.1 18, which being deducted from 25 (the perpetuity), 

 the remainder, or 13.882, multiplied into ij (the given an- 

 nuity), 



