[ 409 ] 



In the fecond produd, we may obferve that there are but two 

 different fadors. That the fador x + m mufl be equal to the 

 fador X _ ^ , or that m is equal to - ' will appear if it be (hewn 



that 



q' a ^- b — c — d 



or 



ab — cd = ■ which may be done 



4 4 ^ 



in the following manner : — Since ah ^ cd — o, we alfo have by 



multiplication, a^ h -\- acd= o, ab- -}- bcd= o, abc j^ c'^ d = o, and 

 abd + cd^ = 0. 



By comparing thefe four equations feverally with the equation, 

 abc + abd + acd + bed = o, we find, 



a'' = ac -\- ad + cd 

 b' = be -{- bd + cd 

 c^ = ab + <7c + be 

 d' = ab + ad + bd 



a"- + b' + c^ + d'' = 2 ab + 2 ac + 2 ad + 2 be + 2 bd+2cd. And 



therefore a + b — e — di' = ^ ab -\- i^ cd •: ■ —ab — cd—o. 



4 

 Qj E. D. 



That the number of different fadors in both produds can- 

 not be lefs than fix, without introducing furds, may be fhewn 

 as follows : 



Vol. VI. 3 F First, 



