( 9* )

§. XXI.

cient-. x. That the Denominator os- each Cocfficient is always a.

3 . That the first Member of cach Numerator, is always a Coefficient of theSeries gy -4* by y ■+ i y 3, &c. vi^. the first Numerator begins with the firstCoelficient g, the Second Numerator with the Sccond Coefficient and seon- 4 * That in ev’ry Member aster the first, the Sum of the Expo-nents of the Capital Letters, is always equal to the Index of the Powerto which this Member belongs : Thus conlidering the Coetficient

k—bBB — zb AC — 3 cAAB — d A+ , . , , ,

-, which belongs to the Power>4,

/1

. we (hall sce that in every Member b B B, x b A C, 3 c A A B, d A 4 , thethe Sum of the Exponents of the Capital Letters is 4 , (where I must takenotice, that by the Exponent of a Letter; I mcan the Nuntber which ex-presses what place it has in the Alphabet ; thus 3 . is the Exponent of the Let-ter D) hence I derive this Rule for finding the Capital Letters of all theMembcrs that belong to any Power; Combine the Capital Letters as osten asyou can make the Sum of their Exponents equal to the Index of the Powerto which they belong. 5 . That the Exponents of the finali Letters, which.are written before the Capitals, expreß how many Capitals there is in eachMember. 6 . That the Numerical Figures or Unicx that occur in thefe Mem-bers, expreß the Number of Permutations which the Capital Letters of ev’ryMember are capable of.

For the Dcmonfiratioti of this ; luppose Ay -h By y -+ C y 3 -f- D y 4,

. &c. Substitute this Series in the room of and the Powers of this Series,in the room of the Powers of there will arise a new Series; then take theCo-eificients which belong to the several Powers of y in this new Series, andmalte them equal to the correlponding Co-efficients of the Series gy ~+ hy y-H i y 3, See. and the Co-efficients A,B,C, D, &c. will be found fisch asl havedetermined thc-m.

But if any one desires to be fiitisfyed, that the Law by which the Coeffici-ents areinform’d, will always hold, fll delire ’cm to have rccourse to theTheorem I have given for Rjisnig an Infinite Series to any Power , or extractingany Root of the fame ; for it they make ufe of ic, for taking succellively thePowers of Ay -+ By y Cy \ &c. they will lee that it must of necellity beso. I might have made the Theorem I give here, much more general thanit is; for I might have fiippos’d, a~ x m -P -i- c 2 See. ~-gy m

-h b y m ~*~ l -+ iy m ~*- 2 See. then all the Powers of the Series Ay -4- B y yC y 3, &c. designed by the universal Indices, must have been taken luc-celfively ; but those who will pleale to try this, may casily do it, by meansof the Theorem for raising an Infinite Series to any Power, &c.

' This Theorem may be applyed to what is called the Reversion of Series,such as finding the Number from its Logarithm given ; the Sine front theArch; the Ordinate of an Ellipse from anArea given to be cut from any Ipoirrtin the Axis.- But to make a particular -Application of it, Pll suppose we have

this