Mif rell anea Curio fa. 195 



tions which the Capital Letters of every 

 Member are capable of. 



For the Demonftration of this •, fuppofe 

 z„—Ay-\-Byy--\-Cy>-\-Dy*, &c. Subftitute this 

 Series in the room of and the Powers of 

 this Series, in the room of the Powers of z, ; 

 there will arife a new Series j then take the 

 Coefficients which belong to the feveral 

 Powers of y\ in this new Series, and make 

 them equal to the correfponding Coefficients 

 of the Series gy-\-hyy-Uy\ &c. and the Co- 

 efficients A, B, C, £>, &c. will be found fuch 

 as I have determin'd them. 



But if any one defires to be fatisfied, that 

 the Law by which the Coefficients are form'd, 

 will always hold, I'll defire 'em to have re- 

 courfe to the Theorem I have given for rai- 

 fmg an infinite Series to any Power, or ex- 

 trading any Root of the lame y for if they 

 make ufe of it, for taking fucceffively the 

 Powers of Ay-X-ByyA^Cy* , &c. they will fee 

 that it muft of neceffity be fo. I might have 

 made the Theorem I give here, much more 

 General than it is \ for I might have fuppos'd, 

 M^'b^+czr^ 2 &c~gy m A-hy m -\~ l \-iy m -\- 2 

 &c. then all the Powers or the Series Ay<~\~< 

 Byy-\-Cy 3 y tkc. delign'd by the univerfal Indices, 

 muft Have been taken fucceffively *, but thofe 

 who will pleafe to try this, may eafily do it, 

 by means of the Theorem for raifing an infinite 

 Series to any Power, &c. 



This Theorem may be applied to what is 

 called the Reverfion of Series, fuch as find- 

 ing the Number from its Logarithm given ; 

 the Sine from the Arc-, the Ordinate of an 



O Ellipfe 



