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one produ6l fhould have all its factors equal to each other ; 

 the other, two of the three fadtors equal to each other. Let 

 x^ + 3^.v' + ;^a'x + a' (= ;v + aj') reprefent one of the produds, 

 and .v^ 'J_ ^^ x' ^ , x -\- b-c {= .v^+T]' X x + c) the other. 

 Then fince thefe produds ought to difter in their loweft terms 

 only, we have 3^2 = 2/* + c, and b' -{■ zbc =1 i,a- , by comparing thefe 

 equations with each other we find ^b^ -{- ^bc -\- c'' =^ ^b^ -\- 6bc 

 :■ b' — zbc — J-' and b = c ; fo that thefe produds will be both 

 cubes, or can have but two different fadors ; but fince 3^ = 2b -\- r, 

 i. e. (from what has been already proved) 5« = 3^, we have 

 a — b ; by which it appears that not only the fame fador en- 

 ters both produds (which has been already fhewn to be ufelefs) 

 but both produds confift entirely of the fame fador equally re- 

 peated and are in efied one and the fame produd. 



In the application of the produds x' — 3;c — 2 and x^ — 3^+ 2, 



to the conftrudion of logarithms the fradion - of the feries 



s 



mufl: be made equal to 



2xX + l]- X x-z + 4 2x ^— i]> x-VH- 2— 4 



or in its loweft terms ^ + i ^ x x~z orlv^' X ;c + 2 



Thi 



