Vol. XL.] PHILOSOPHICAL TRANSACTIONS^ 273 



«» = 1/ aa — b = 22 ; therefore y = 25 — 22 = 3 ; and hence the binomial 

 reduced is 5 + ^/3. 



Demonstration — Take any binomial, as x/a-]-\/b, which suppose reducible 

 to the binomial x -\- '/y; then, by cubing, x' + 3xx^/y -\- 3xi/ -{- yVy =. 

 a + ^b. 



Put x^ + 3x7/ = a, and 3xx>/y + y^y = ■/i. 



Then whatever the index of radicality may be, from the square of the former 

 part subtract the square of the latter, and there will remain x^ — 3x*y + 

 3xxyy — 1^ ■=. aa — b\ then extract the nth root of both sides, that is, in the 

 present case the cube root, it will give xx — y ■=z >J aa — b; or making 

 is/ aa — b = m, it will he xx — y ■=. m, and therefore y = xx — m. Now 

 in the above equation writing a for a:^ + Zxy, and xx — m for y, there results 

 the equation 4a^ — Smx = a. 



Now resume the equation 2x = »/ a -\- ^ b + \/ a — ></ b % and to take 

 away the radicality ^, make a -\- ^b ^ z^, and a — »/b ■=■ v^, there will 

 then result these two new equations, z* + t;* = 2a, and i -\- v =. ix ; it fol- 

 lows therefore that—,— = -. But — , — = z'* — zf + w^ = -; and besides 



Z + DT Z + V * 



zz + ^zu •\- w ■=. Axx. 



Taking the difference of these equations, there results Szu = Axx — - ; but 



X 



zV -z:^ aa — b \ therefore zv ■=■ V aa — b, which being put = m, there will 



arise 3m ■=■ Axx , or A3? — Zmx = a, which is the very same equation 



that came out before ; and thus it will revert to the same in every case of radi- 

 cality whatever. 



To try therefore whether the expression "V a -\- »/ b can be reduced to a sim- 

 pler state; put 2x = ^/ a + v^i -f 1/ a — »/ b, and ^aa — Z; = m, also y = 

 XX — m; then the expression reduced will be a? + ^y, if it can admit of in- 

 tegral, or at least rational quantities. 



But in case these should not be integer or rational quantities, yet the rule 

 above will be of use in solving equations of any kind, as will be seen here- 

 after. 



In the mean time perhaps this doubt may arise, whether this rule will obtain 

 universally in any powers whatever of the binomial, viz. whether in any bino- 

 mial, whose index is n, if from the square of the sum of the terms in the un- 

 even places, there be subtracted the square of the sum of those in the even 

 places, the remainder will be another binomial having the index n. To which 

 Mr. Demoivre answers, that this is a fact which has been observed by many 



VOL. VIII. N N 



