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ILGORITltDl OF IMAGINARY QUANTITIES. 



imaginary 

 quantJties, 



sake of siroplicity, we will represent by a and 6, the third 

 root being 1. Then from the known theory of equations 

 we have I -\- a -\- b =10, or a-\- b~ — 1; 



also 1 X a X bzz ab zz 1 ; 



Answer to the Now we have also a' =r 1 ; ^' — 1 ; but there is this differ- 

 question on pjjce with regard to these products represented by 1 ; viz. 

 that the first has no limitation as to its roots, being the pro- 

 duct of different factors, and is therefore as general as 1 

 unconditionally assumed ; whereas the two latter have only 

 particular roots, the first being «, and the second b, and 

 they can have no other roots ; if, therefore, in the operation 

 of squaring, we have to take the roots of these quantities, 

 we must pay particular attention to this circumstance ; the 

 want of which, in the numerical example in my former let- 

 ter, is what gave rise to the incongruous result there de- 

 duced. Let us now square this literal expression by the 

 commoB method, only observing, with regard to the products 

 end powers, the rules above laid down : 



Va ■{- Vb 

 Va-\'Vb 



'/a* + Vab 



+ Vab + a/ ft* 



Va* + 2Va 5 + ^ ft* = 



.4Lnd that this is the true result, may be shown as follows. 

 The p roposed formula 



»/ — i + iV— 3+ V^—l — 



y'— ,3, or 



is the real root of the equation 



asdeduced from Cardan's rule ; and consequently the cube of 

 that formula, minus three times itself, ought to be equal to 

 •^1. Let us therefore continue the operation, and see how 



far 



