( ipO 
a Letter ; I mean the Number which expreffes what Place it 
has in the Alphabet ; thus 4 is the Exponent of the Letter D) 
hence I derive this Rule for finding the Capital Letters of all 
the Members that belong to any Power 5 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 Ex- 
ponents of the fmall Lecters^which are written before theCa- 
pitals, exprefs how many Capitab there is in each Member, 
6. That the Numerical Figures or Uncia that occur in thefe 
Members, exprefs the Number of Permutations which the 
Capital Letters of every Member are capable of. 
For the Demonftration of this ; fuppole 2s= Byy-^ 
D)\ &c, Subffitute this Series in the room of z.^ 
and the Powers of this Series, in the room of the Powers of 
^ \ there will arife a new Series 5 then take the Co-efficients 
which belong to the feveral Powers of in this new Series, 
and make them equal to the correfponding Co-efficients of 
the Series gy hyy'\^ if^ &c. and the Co-efficients A^ B^ 
P, &c. will be found fuch as I have determined them. 
Bui if any one defires to be fatisfied, that the Law by which 
the Co-efficients are form'd, will always hold, I'll defire 'em 
to have recourfe to the Theorem I have given for raifiog an 
infinite Series to any Power, or extra<^ing any Root oi the 
fame ; for if they make ufe of it, for taking fucceilively the 
Powers of Ay -j- Byy -^- Cy^^ &c, they will fee that it rnufl 
of neceffity be lo, I might have made the Theorem I give, 
here, much more general than it is 5 for I might have fop- 
pos'd, 
az'^^ hzj"^' + czT^'' &c.= gf-}- iy"^ &c. then 
all the Powers of the Series Byy\' Cy\ &c. defign'd 
by the univerfal Indices, mult have been taken fucce{EveIy ; 
but thofe who will pleafe to try this, my eafily do it, by 
means of the Theorem for raifing an infinite Series to any Towers 
&c. 
This Theorem may be applied to what is called the Rever» 
fion of Series, fuch as finding the Number from its Logarithm 
given ; the Sine from the Arc ; the Ordinate of an Elliple 
from an Area given to be cut From any^ Point in the Axis i 
But to make a particular Application of it, I'll fuppofe we 
have this Problem to folve i ifiz^ The Chord of an Are be-- 
