ALGORITHM OF IMAGINARY QCAMTITIKS. 197 



far the result will a^ree with the required conditions: Answer to tha 

 for this purpose we will repeat again the preceding square, 'oi^on 



I 333 



imaginary 

 quantities. 



4/0' + i/« + -/aft* 



33 3 I 



a 4- 3 ya + 3 ./6 + 6 = (vc + </^)' 



This result is evidently deduced according to the pre- 



3 3 



ceding rules, in which it is shown, that v'a' =: o, -v/i' =: 6, 



3 » - 33 



V'aft =: v'l rr 1, and consequently v'a*6 r: y'ada zz 

 3 J 3 3 



y'« and t/ah^ z=. >yah'h r: v'6 



We have seen also, that, o and & being the imaginary 

 roots of the equation x^ zz 1, a + ir: — 1; the above 

 expression may therefore be written 



jr» = (^fl + ^&) = 3 (v'a + '/&) — 1 



3 3 3 3 



also — 3 X r: 3 (y^a + v^6) = 3 (-/a + -/6) 



whence x' — 3 x rr *— 1 



as it ought to be ; which shows, that our operations have 



been accurately performed. 



It only remains, therefore, now to explain io what respect 

 our numericaloperation differs from that which has just been 

 performed upon the literal symbols a and b. Now, upon a 

 comparison of the two, we shall find them to be precisely the 

 same, except, that in the former we have not retained the 

 square form under the radicals; by which we lose sight of 

 one of the particular circumstances, which ought to be kept 

 in view throughout the operation. It happens m the exam- 

 ple, proposed, that the imaginary quantities under the ra- 

 dicals are powers of each other; as is the case with the 

 roots of every binomial equation x" 1= 1 ; that is, in our 

 particular examples, a* zzh and 6*r: a\ but this equality 

 has not place when the (^^aantitifts enter under a radical, and 



the 



