VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 275 



Now putting r = 1, for brevity, then the general form of these is 

 2- Xaf- 2-' X ^x- + 2- X J . ^^-^ - 2-' X » !i=l. Izf^-V 



1 12 X 2 3 



&c. ^ c. 



The difference in these equations consi sts chiefl y in this , (hat the former are 

 derived from the equation 2x = 1/ a + v'^ + "V a — ^h, but the latter from 

 the equation 1x =:. 1/ a -\- */ — b — V a — n/ — b; and if this latter equation 

 be freed from its general radicality, there will be obtained equations for the 

 cosines. 



Let therefore the equation 1x =. \/a-\-'s/— b-\-\/a — »/~b be pro- 

 posed to be freed from its radical sign ]/ . 



Put \/ a + ^ — 5 = z, and Va —\/ — b =■ v ; also put z -^ v ■=. 1x. 

 Hence it will be 1st, z* = a -f y^— i, and 2nd, \?=.a—i^—b; conse- 

 quently 7? -\- %? ■=. la. But since z + w =: 1x, therefore ^ ^ = - =: zz — 



zv -f- w. But (z + 'vf' = zz -|- 2zi; + w = Axx; therefore Zzv = Axx — -. 

 Now since zV = wa -}- i ; therefore zv = 1/ aa ■\- b; which being put = m, 

 it will then be Axx — - = 3m, or Ax^ — Smar = a. 



X ' 



Hitherto we have had two kinds of equations ; the first, in which m was 

 put = ^/ aa — b ; the latter, in which it was ^ l/aa -{■ b. The former may 

 be called hyperbolical, the latter circular. 



Prob. 2. — To extract the Cubic Root of the Impossible Binomial a -\- s^ — b. 

 Suppose that root to be x + v' — y, the cube of which is x^ -j- 3xx\^ — y 



— ^^y — yV—y- 



Now put a^ — 3xy = a, and 3xx»/ — y — yi/ — y ■=■»/— b. Then the 

 squares of these will give two new equations, viz. 

 x^ — 6T*y + ^x^y"^ = aa, 



— ^x^y + Qx'^y^ ~ xf ■= — b. 



Then the difference of these squares is .r" + 3,r*y + ^x^y* -\- if ■= aa •\- b\ 

 the cubic root of which \^ xx -\- y =. s/ aa -^ b •=■ m suppose ; hence y = m 

 — XX ; which value of ^ substituted in the equation o^ — 3xy = a, gives x^ — 

 Smx -j- 3x^ = a, or 4x^ — 3mx =. a ; which is the very same equation, as 

 had before been deduced from the equation 2x = 1/ a -\-'/ — b -^-V a —»/ —b\ 

 but yet it does not follow, that in the equation Ao^ — 3mx = a, the value of ar 

 can be found by the former equation, since it consists of two parts each in- 

 cluding the imaginary quantity ^/ — b ; but this will be best done by means of 

 a table of sines. 



Therefore let the cube root be extracted of the binomial 81 -f i/— 2700. 

 Put a = Ql, b = 2700: then aa + b = 656i + 2700 = 9261, the cubic 



N N 2 



