276 PHILOSOPHICAL TRANSACTIONS. [aNNO 1698. 



combine the capital letters as often as you can make the sum of their exponents 

 equal to the index of the power to which they belong. 5. That the exponents 

 of the small letters^ v.fhich are written before the capitals, express how many 

 capitals there is in each member. 6. That the numerical fio-ures or unciae that 

 occur in these members, express the number of permutations which the capital 

 letters of every member are capable of. 



For the demonstration of this ; suppose z = a?/ + ^yy -(- Cj/' + ny* &c. 

 Substitute this series instead of z, and the powers of this series instead of the 

 powers of x ; there will arise a new series ; then take the coefficients which be- 

 long to the several powers of y, in this new series, and make them equal to the 

 corresponding coefficients of the series gy -)- hyy + iy^ &c. and the coefficients 

 A, B, c, D, &c. will be found such as I have determined them. 



But if any one desires to be satisfied, that the law by which the coefficients 

 are formed, will always hold, they may have recourse to the theorem I have 

 given for raising an infinite series to any power, or extracting any root of the 

 same ; for if they make use of it, for taking successively the powers of Ky + 

 ^yV + ^y* ^^' ^^^y ^''' ^^^ ^^'^^ ^^ must of necessity be so. 



I might have made the theorem I give here, much more general than it is ; 

 for I might have supposed, «»"' + ^z™ + • + cz™ + ^ gjc. = gy"" -\- hy"" + • + 23/™ + 2 

 &c. then all the powers of the series Ay + hyy + Cj/' &c. designed by the uni- 

 versal indices, must have been taken successively ; but those who will please to 

 try this, may easily do it, by means of the theorem for raising an infinite series 

 to any power, &c. 



This theorem may be applied to what is called the reversion of series ; such as 

 finding the number from its logarithm given ; the sine from the arc ; the ordi- 

 nate of an ellipse from an area given to be cut from any point in the axis : but 

 to make a particular application of it, I will suppose we have this problem to 

 solve ; viz. The chord of an arc being given, to find the chord of another arc, 

 that shall be to the first as w to 1. Let y be the chord given, z the chord re- 

 quired ; now the arc belonging to the chord ^ is, ?/ -}- -jr-rr + "loTT + 

 -^^ &c. and the arc belonging to the chord z is z -f- -^- -|- -£7- -j- — ^ &c. 



the first of these arcs is to the second as 1 to w ; therefore multiplying the 

 extremes and means together, we shall have this equation : 



;ri 

 find a = 1, i = 0, c = -j^, f/ = O, e = -^ /= o, &c. g = n, k = O, 



&c. 



Compare these two series with the two series of the theorem, and you will 



— , a — 0,e— ^y^^ 



