[ 407 ] 



term of thit prodnd. Suppofe one of the fadors tohe x + m ; the 



+? a - '^ 



Other will be a: - m. Then qm—m'— r -.■ m — -1- + ^^~- -r, and 



m 



-+ ^— _r Let the fadors of the produd x* -{- qx^ -{- 



4 



rx- + sx -\- 1 ht X + a^ X -[- b, x -^ c, and x -{• d. And the fadors 

 of the produd x'' -{-qx -j^ r will be, x -\- II — Z_ Ji 



y/: — — — ~ — Z — ' — ab — ac — ad — be — bd — cd, or (which is 



^1, r ^u- \ , a^ b -\-c -^ d + a^b — c — dV 



the fame thing) x\ v. ^^_^_a^ _ ^^ _ ^^_ 



2 4 



Thefe fadors will be always rational (whatever may be the 



values of a, b, c, and d, provided they be rational) if we make 



<3 + cd= 0. Then fince j, or abc + abd + acd-^bcd= o, we have 



— (= — a) = --_ _. And dividing by cd and multi- 



b be -\- bd -|- ed 



plying by the denominators, be -^ bd -\- cd — b' . Therefore 



b^—bd b^—bc . c-\-d 7~r2V 



c= ,-^-^-' d = -^-- and b = --± v. L±^' ^ cd So 



cd 

 that if b be rational, c, d, and a (or — — ) will alfo be rational. 



b 



But ^ will be rational, if c' + 6<:^+ d' be a perfcd fquare. Make 

 c^ -\-bcd-{-d'=c\ Then6r^+</'=oand^=-6c. Univerfally, putting 

 f ' +- 6cd + d^ = c'" + 2 en + »'', we have 6d — 2«. c = »* —d^ 



and. 



