[ 225 ] 



Let the rationals colleded=^, and transferring to the other 

 fide as well the rationals as the iimplc quadratic ^xs/b, the equa- 

 tion will be 2 »!■ Va + 2 '^ab = 2 >ycd-\- 2 \/c/+ 2 </ df—% x \fb—g. 



For brevity fake I fhall hereafter affume letters for the rationals 

 of the equations, and transfer each furd as it comes out to a fimi- 

 lar one wherever it is found, annexing the fign according to the 

 (ide it is placed at ; fquaring the above equation we have 



—Agx'^b —4gs/df-4gVcf +SfVcd 



Here then we have an equation free from the furd \/a and fincc 



the coeifficients of the furds can by no means afFed the poflibility 



of -clearing it, and we arc fo-far only concerned, I fhall omit them, 



and. fubftituting k for the rationals /' — b, the equation will be 



•/b + \^bcd+ >Jbcf-\- i/bdf = Vdf+ sfcf-V s/cd — k and fquaring. 



/+ lb s/cd+ 2b s/€j-\- %b >/df =m + 2f>Jcd^2d-/fc—ik Vdf 



+ 2bf^/cd + ibd >/cf+ 2bc s/df —tk -^cd-zk s/fc + ic -/df, 



whence negleding coefficients after uniting the furds ; 



we have -Jcd-V -Jcf — -t/df^n. Here we have exterminated the 



furd 'Jb, and by each fuccecding operation we take away one more, 



always adhering to the rules to transfer to the fame fide all thofe 



furds whofc fuffixed quantities involve the fame letter ; from 



whence alfo it follows that the number of involutions will be plus 



one than the number of furds. In the fecond ftep there were 



eleven furds with rationals, and in the next tho' twelve appear; 



Vol. VI. F f they 



