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By fquaring the fides of the equation propofed, we will have. 



If all the combinations that can be made out of the trinomial 

 >/c + s/d + s/f be obferved, which are only three, we will fee, that 

 the fuffixed quantities of every two of thefc admit one common 

 divifor ; place then any two of thcfe combinations at the fame fide, 

 and the reft of the terms at the other, and afiTuming the rational 

 quantities colleded —p we have, 



p-\- zs/ab — 2 Vcd=z\/cJ -\- i*/df^ and by fquaring 



p-'-^- \ab-^r ifCd-\- i,p s/ab-\p s/td—%-Jabcd—\ cf+ 4 df+ Sf\/dc. 



Of thefe furd rccStangles, it will appear, that two coalcfce, i. e. 

 have the fame fuffixed quantity, and the fuffixed quantities of the 

 others admit a common divifor. By placing the rationals colleded 

 p^ -\- j^ab-\- Jii-dc -^ cf—\ df= y, and the coefficient if—^p = r, we 

 have 4/ t/ab—^ s/abcd=r</dc—q^ and fquaring, 



i6j!>' ab-\- 64 abcd—(n\.pab s/cd= r^dc + q'^—zqr s/cd: now putting 

 the rationals =yj and the coefficient, (i\pab—% qr — t^ the equation 

 will ftand, f= t ^cd, and fquaring,/" * = fed, which latter equation 

 is free from furds. It is plain that aflliming letters for the rationals, 

 is only for brevity, and no way affeds the furds. Thus it will ap- 

 pear, how an equation, involving five quadratic furds, is rendered 

 rational, by the fimple rules of involution and tranfpofition. 



Havinc 



