377 



equally to two or more roots which agree in their leading figures to the 

 extent mentioned. These are equal roots of an approximate equation, 

 and have the same claim to be considered equal roots of the proposed 

 equation as either of them has to be considered a root of it. The con- 

 clusion is the same, whether at a more remote figure the roots would 

 separate, or the condition of imaginary roots, hitherto delayed, be after- 

 wards fulfilled for a final triad. In either case a real value is found 

 which satisfies an equation so nearly coincident with the equation pro- 

 posed that, since approximations only are attainable, the two equations 

 may be regarded as identical — in so far, at least, as the roots thus far 

 common to both are concerned. 



No doubt, in thus prosecuting the development of one of a pair of 

 contiguous roots, there may be abiding uncertainty as to whether the 

 roots are strictly equal or not ; for the decimals being interminable, all 

 that we can afiirm, however far these decimals are carried, is, that thus 

 far, at least, the roots are undistinguishable from equal roots — supposing, 

 that is, that a separation has not yet taken place. But in the case of a 

 pair of imaginary roots, there need never be ahiding uncertainty at all. 

 The approximative process may fail to separate a pair of real roots — for 

 they may not be separable ; but a pair of imaginary roots must, sooner 

 or later, unfold their character as such, by means of the tests of ima- 

 ginarity. There are no such tests for incommensurable equal roots : 

 by continuous approximation, f(x) may, in the one case, continuously 

 tend to zero interminably ; in the other case, it cannot approach zero 

 within a certain finite limit ;* but, in either case, the process may be 

 stopped when a sufficient number of decimals is obtained; and the 

 inexact root, thus far developed, will be an exact root of an equation so 

 nearly coincident with that proposed, that it may be substituted for it 

 without appreciable error, in so far as concerns the particular roots in 

 question. 



It should be observed, however, that the approximate equation, of 

 which a partially developed root of the proposed equation is an exact 

 root, is not precisely the same as the approximate equation for another 

 partially developed root ; since for different inexact roots the correction 

 for the absolute number of the proposed equation will most likely be 

 different ; and similarly as respects different groups of approximate equal 

 roots. But unless the correction in each case be so small as to be of no 

 practical consequence, the approximate equation will not have ap- 

 proached near enough to the proposed to justify its being substituted for 

 it. All that the present Note affirms is, that a pair of real roots may 

 have so many leading figures in common, or a pair of imaginary roots 

 may have the imaginary element so insignificant, that both in the one 

 case and in the other the roots may be regarded as real and equal- 

 satisfying the conditions of the equation with as much precision as the 



* If it be not the final triad, but a preceding triad, which indicates the imaginarity, 

 then, as the approximation will not be directed to a root of/i(» = 0, the successive 

 values of /(a?) may even diverge from zero, though not beyond a finite limit. 



