1 86 Mtfcellanea Curiofa* 



Demonjlration. 



To raife the Series az.-\-bz,z,+cz.* + ak. 4 ,&c. 

 to any Power whatfoever, write fo many Se- 

 ries equal to it as there are Vnits in the In- 

 dex of the Power demanded. Now it is evi- 

 dent that when thefe Series are fo multipli- 

 ed, there are feveral Products in which there 

 is the fame Power of z> 9 thus if the Series 

 az, + cx J + ^, 4 , &c. is rais'd to its 



Cube,, you have the Products b 3 z. 6 , abcz, 6 ^ 

 aadz, 6 , in which you find the fame Power z, 6 

 Therefore let us confider what is the Condi- 

 tion that can make fome Produces to contain 

 the fame Power of £, the firfl thing that will 

 appear in relation to it, is that in any Pro- 

 duel: whatfoever, the Index of z. is the Sum 

 of the particular Indices of z, in the multi- 

 plying Terrris (this follows from the Nature 

 of Indices) thus b 3 z. 6 is the Product of bz, 2 , 



.b£\ bz?, and the Sum of the Indices in the 

 multiplying Terms, is 2" \ + 2 + 2 6 ; abcz. 6 

 in the Producl of az.^ bz,z, czJ , and the Sum 

 of them Indices of £ in the multiplying 

 Terms is 1 * x 2^326 aadz. 6 is the Product 

 of az,^ dz,^ dz,*, and the Sum of the Indices 

 of % iti the multiplying Terms is 1 + 1+4=: 6\ 

 the next thing that appears is, that the In- 

 dex of S, in the multiplying Terms is the 

 fame with the Exponent of the Letter to 

 to which z, is join'd, from which two Confi- 

 derations it follows, that, To have all the Pro- 

 ducts belonging to a certain Power of z, you mufi 



■find all the Products where the Sum of the Ex- 



. fonents 



