square, 



EQUATION. 



The values of the cube roots of a3 are a 



- - a, - 

 and the values of the cube root of ti are 6, 



3 / r A3 J7~? Hence it appears, that there are nine 



^ ""3* V r~~27-' Also since 3 values of a + 6, three only of which can 



. - answer the conditions of the equation, the 



-4- 63 4- r = b> = -t I 3-1 others having been introduced by involu- 



2 V4 27' tion. These nine values are, 



and6== /--^ 



= 7 - 

 V 2 



27 



We may observe that when the sign of 



J 7 ~~l^ in one P art of the expression, 

 v 4 27 



is positive, it is negative in the other, that 



Since b = -, the value of a: is also 

 3a 



3 1 r T = r == * n *he operation we assume 3 a b = 17, 



.J ^ / ^ _ 3 i that is, the product of the corresponding 



^ 4 27 values of a and b is supposed to be possi- 



ble. This consideration excludes the 2 d ' 



3 3 /_ r 1 71 g 3 3d. 4th. 5th. 7th. and 9th. values of a 4- b, or 



V 2 V 4" ~yf' oc ; therefore the three roots of the equa- 



ton are 

 Ex. Let x3 + 6 # ->- 20 = ; here 



- .. 



2 



Cor. 1. Having obtained one value o^ 



x, the equation may be depressed to a Con 3> This solution on , extendg t 



quadratic, and the other roots found. those cases in which the cubic has two 



, impossible roots. 

 Cor. 2. The possible values of a and 



b being discovered, the other roots are 



known without the solution of a quadra- For if the roots be m 4- ^n,m v/Tn, 



tic. and 2. m, then q (the sum of the 



