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VOL. XIX.3 PHILOSOPHICAL TRANSACTIONS. 177 



+ - X^- X -J- X -j-a bz 



+ - X — — a bd 



+ ni , . m — 1 m - 2 2 

 - X a c 



1 2 



+ -a e 



+ ^ X '^' X '-^' X -^^ X ^;-V-^ iV"* = &c. 



4. ^ X "-^^ X '"-^ X '!i^a"'-* ^3, 



For the understanding of it, it is only necessary to consider all the terms by 

 which the same power of z is multiplied ; in order to which, I distinguish two 

 things in each of these terms : ] . The product of certain powers of the quan- 

 tities, a, b, c, d, &c. 2. The uncias, as Oughtred calls them, prefixed to these 

 products. To find all the products belonging to the same power of z, to that 

 product, for instance, whose index is m + r, where r may denote any integer 

 number, I divide these products into several classes ; those which immediately 

 after some certain power of a (by which all these products begin) have b, I call 

 products of the first class ; for example a™ - *b'^e is a product ol the first class, 

 because b immediately follows a™ - * ; those which immed-ately after some 

 power of a have c, I call products of the second class, so a'" - ^ccd is a product 

 of the second class; those which immediately after some power of a have^, I 

 call products of the third class, and so of the rest. 



This done, I multiply all the products belonging to z™ + »•-', which pre- 

 cedes immediately z'" + % by /', and divide them all by a ; 2. I multiply by c 



VOL. IV. A A 



