440 On Imaginary Cube Roots. [June, 



By means of these roots, and 28 varieties connected with them, 

 the cube roots of all binomials may be obtained, if such roots admit 

 OS a finite expression, even when they are irrational, and without 

 trial or assumption. 



The imaginary quantity which I introduced relates only to t, and 

 to the second cube root in the column on the left hand, which cube 

 root is thus demonstrated to be exact : — 



Letl t — P t= c 



"~2 2 



subtracting — internally to the parenthesis 



o.(._-L)w(f-|)w(4-^) 



adding — 5 — to one side of the equation, and its equal — to the 

 other side. 



extracting the cube roots 



No other value can be used in this case for the cube root of the 

 binomial, which the algebraist may readily prove by adapting it to 

 an irreducible equation where tliere is no ambiguity in respect of the 

 square root. Such is the equation x^ — 6S x = 16"2, where the 

 binomial is 81 d- V — ^JOO, and the cube root for i is — 3 + 

 a/ — 12. 



To obtain the imaginary quantity which is the subject of consi- 

 deration, I employed the reducible equation a;' — 24 a; =: 72, 

 where X = 6, t = S + ^ — 3, w = 3 — \/ — 3 ; and by 

 Cardan's rule the roots of the equation are thus expressed : — 



\/ 36 + V 784 + ^ 36 - \/T84 



and by the previous demonstration, the cube root of the binomial on 

 f v fr liHud connected with / is the quantity I gave; namely, 



-g + a/ - (^-y - "T V - 3j. 



<ts universally give precedency in magnitude to the 

 the left handj and in this they follow the old masters, 

 3 



