MifceUaneaCuriofa, 1 89 



generally, You mufi prefix to any ProduEl where" 

 of any Power of the infinite Series azH~bzz~i~ 

 CZ 3 ,&C. is composed the Number which expref- 

 fes how many ways the Letters of each TroduEh 

 may be changed. 



Now to find how many ways the Letters 



of any Product, for inftances a m ~~ n b h c$ d r 

 may be changed \ this is the Rule which is 

 commonly given : Write as many Terms of 

 the Series 1x2x3x4x5, &c. as there are 

 Units in the Sum of the Indices, 'viz. *»•— # 

 T ^ "It p 4- let this Series be the Numera- 

 tor of a Fraction whofe Denominator fhall 

 be the Product of the Series 1x2x3x4x5, 

 &c. 1 x 2 x 3 x 4X 5,&c. ix2X3X 4x5x6, 

 &c. 1x2x3x4x5, &c. whereof the firft is 

 to contain as many Terms, as there are 

 Units in the firft Index m^n, the fQeond 

 as many as there are Units in the fecond In- 

 dex h ; the third as many as there are Units 

 in the third Index p ; the fourth as many as 

 there are Units in the fourth Indexs r. But 

 the Numerator and Denominator of this 

 Fra&ion have a common Divifor, vtil the 

 Series 1x2x3x4x5, &c. continued to fo 

 many Terms as there are Units in the firft 

 Index m »— n \ therefore let both this Nume- 

 rator and Denominator be divided by this 

 common Divifor, then this new Numerator 

 will begin with ^^«-hi, whereas t'other 

 began with 1, and will contain fo many Terms 

 as there are Units in h + p^r, that is, fo 

 many as there are Units in the Sum of all 

 the Indices, excepting the firft \ as for the 

 new Denominator, it will be the Product of 



three 



