[ 4^o ] 



First, neither produd can have three fadors ahke — Since, by 

 a change in all the fecond terms, any fador may be reduced to 

 a fimple expreffion ; let the fador which occurs three times be 

 denoted by x, the produd will then be deficient in the three 

 lowcft terms — tlie other produd that it may differ from this by its 

 loweft term only, muft want the penultimate and antepenulti- 

 mate terms ; at the fame time, retaining the laft ; fo that two of 

 its fadors will be impoffible. 



The above is evidently applicable to produds of all dimcn- 

 lions ; and from this immediately follows what has been proved 

 above, relative to the Icafl number of different fadors in produds 

 of three dimcnfions. 



Hence it appears that the leaft poffible number of different 

 fadors will not be lefs than the index of the higheft term in 

 either produd, if that index be even — if it be odd, the leaft 

 poffible number of different produds will be greater than the 

 index. 



Whence it follows that the number of different fadors in the 

 prefent cafe cannot poffibly be Icfs than four. 



But in the next place, the number of different fadors cannot 

 without introducing furds, be lefs than fix. 



For 



