VOL. XXV.] PHILOSOPHICAL TRANSACTIONS. 33^ 



Example 2. — In the equation ** = 20^7' + Ibloc^ — 6592^ + 21312, it 

 will be /> = 5, y = i76, r = — 384, s = 13072. Hence p^ + 9 = 201, 

 and 2/)r + ^ = 9232, andr^ = 147456; and thence a® = 201a* •— 9232a^ + 

 147456. Now in the cubic theorem it will be /> = 67 ^ and q = *V*> ^"^ 

 r == 65219; ^n<^ ^he cubic root of the binomial 65219 + y' 3 s s s 9_y 707. ^jH 

 be V 4- i/ VV- Therefore a^ = 67 + 77 = 144, or a = 12 ; and hence 

 j7 = 5 — 12 ± ^ lb + 176 -h 144 + 64 = — 7 + 11 = 4 or — 18, and 

 j7 = 5 + 12 ± V25 + 176 — 144 — 64 = 17 + -v/ —7, two impossible 

 roots. 



Now the investigation of this theorem is as follows. By multiplying to- 

 gether the two quadratic equations z* + laz — b = O, and z* — 2az — c = O, 

 is formed the biquadratic equation z* = 4a* -{- b -\- c .z* + lac — 'lab . z — bcj 

 where the second term is wanting, and which put equal to the equation 

 z* = ez* -{• fz -\- g. Hence, first, 4a* + ^ + c = e, or /; = e — 4a^ — c. 

 Secondly, 2ac — 2a^ = f, that is, 2ac — 2ae -f 8a^ + 2ac = f^ hence c = 



(-+-. — 2a", and thence ^, = e — 4a*-^ — ■^ + 2a' = — ^+- — 



2a^ Thirdly, - ^,c = ^, or - ^ + ^* - 2ff«* + 4a* = -g, that is, 



a® = 4^ea* — -^g-a" — -rr^^'^ + tV/*j which is in effect a cubic equation, com- 

 posed of a" and the known or assumed quantities e, f, g. Now that root may 

 be exhibited by the first theorem, and b and c will become known by the same 

 calculation. But the roots of the equations z* + 2az — ^ = 0, and z^ •— 2az 

 — c = 0, are z = — a+ Va^+^j ^i^d z = <z + '^a* + c, orz = — a ± 

 V/- — c^ — -^j and z =r a -l-\/-e — a* -I- — , which therefore will be 



V 2 4a' — V 2 ' 4a» 



the roots of the equation z* = ez* + /z + §"> when a or a* is known from the 

 equation a° = 4-ea* — ^ga^ — iV^^^ + Vi:/*' Now that this equation may be- 

 come general, and furnished with all its terms, make z = a; — />, then will 

 ** •— Apx"^ + 6/) V — 4/)^37 + P^ = ^•«^* —■ ^pejc -f p^e + foo — fp -\- g, also 



« = p — a ± \/^e - a* — £, and^ = p + a ± \/ie - a^ -f £. 



Lastly, for convenience and brevity, make e = 29 + 2p^j andy= 8r; then is 

 ** — 4pa?^ -f- 4p V = 29J?" — 4p^j: -I - 2p^9 + p* + 8ra7 — 8/>r -|- g-, 



X =p ^a ±\/p^ 4- 9 — a* -— , also ^ = p + a lyp* + 9 -- a^ -|--, 



and a® = p* -f- 9 . a* — ^9 + ^.p'^ -j- i.p*^ + ^^* . a* + r^ Finally, make 

 g=z As — q^ -\- 8pr — p* — 2p*y, and the foregoing equations become 

 X* = 4p.r^ + 2qx^ + 6rx -j- 4*, and a^ = p^a* — Ipra^ -|- r^ 

 ^Ap^-^Apq^q-" + ^ - ^ 



X X 2 



