Young — Algebraical Equations of the Third Degree, S^c. 39 



this function to vanish, causes it to do so between like signs, then is 

 rj, with its proper sign, the indicator of a pair of imaginary roots in 

 the contiguous quadratic function, when this is equated to zero. If 

 a value (rs) of a?, which causes this quadratic function to vanish, causes 

 it to vanish between like signs, then is ri, with its proper sign, 

 the indicator of a pair of imaginary roots in the contiguous function 

 of the third degree, when this is equated to zero, and so on ; the 

 sign of ri, r^, &c., denoting the region in which the pair of imaginary 

 roots is — not situated, but indicated. And what, in this case of a pair 

 of imaginary roots, is the indicator, becomes, in the case of a pair of 

 unequal real roots, a separator of those roots. 



A pair of roots indicated in any interval \_a, &], if they turn out 

 to be imaginary, will be equally indicated in the indefinitely narrow 

 interval [r - o, r + ^], be ^ ever so small, r being the indicator. The 

 modification which the coefficients of that quadratic factor of the 

 function to which these imaginary roots are due (in order that the 

 two roots indicated between the above narrow limits may be real 

 roots, the other roots of the function remaining undisturbed) must 

 evidently be such as to make the quadratic factor the complete square, 

 {x - ry.'^' 



Of course, the same pair of equal roots (r, r) would replace the 

 pair of imaginary roots by making a suitable modification of the final 

 term of the function alluded to, without any interference with the 

 other coefficients ; but, then, all the remaining roots of the function 

 would be changed ; since, for no one of the values of x, for which the 

 unaltered function vanishes, would such evanescence take place when 

 the final term, or the absolute number, only, was changed. 



NOTE. 



On the formula [4], in Article (2) 



The general expression [4], for the three roots of a cuhic equation, is entirely 

 free from superfluous values. The values there symbolized are just three in num- 

 ber, the cube root being the only item in the formula which involves multiple 

 values. The symbol r represents one only of the two roots of the quadi-atic equa- 

 tion Q2 _ 2,PP' = ; lohich one of the two is entirely matter of choice. If after the 

 root selected has been employed, the other root be introduced into the formula, in its 

 place, we shall get a second expression for x, differing from the first one only in 

 appearance; and, symbolizing the same three values, the analytical investigation of 

 the formula sufficiently shows such to be the case. 



* The quadratic factor, alluded to above, is that which enters the derived equa- 

 tion of lowest degree, into which imaginarity is transmitted from the primitive 

 equation ; the quadratic factor spoken of belongs to 'Ca& primitive equation only when 

 the imaginarity so enters that it is not transmitted to a derived equation. 



