282 Prof. J. R. Young on the decomposition of Functions 



gree may be expressed in pairs of the forms « + /3, a— /3 ; 

 aj + /3j, «i — /3i, &c. ; and these seem entitled to be called 

 conjugate pairs. This way of pairing the roots of equations 

 has already been distinctly noticed by Professor Davies in a 

 former Number of this Journal. See vol. xxxiii. p. 366. 



In reference to the formulas established in this paper, it 

 may not be superfluous to observe that tliey will be found to 

 be more especially useful in those cases in which all the roots 

 but two are real ; as they will enable us to exhibit the imagi- 

 nary pair, by aid of the real roots, with comparatively little 

 expense of calculation ; and even when all the roots are real, 

 a saving of figures is still effected by them. But, in comparing 

 formulas of this kind with the numerical process of Horner, 

 it must always be remembered that Horner's method supplies 

 the roots in an explicit form ; whereas, in expressions for them 

 such as these, there yet remains an unperformed operation, 

 indicated by the radical ; which, however, in the case of ima- 

 ginary roots, is of course impracticable ; and therefore leaves 

 nothing further to be done. But, in all formulas for imagi- 

 nary roots, into which approximate values only of the real 

 roots enter, it is necessary, in delicate cases, that is in those 

 cases in which a very slight change in any of the coefficients 

 would convert unequal into equal roots — it is necessary, in 

 such cases, to push these approximations to a more than usual 

 extent, in order to avoid the conversion of imaginary roots 

 into real, and vice versa; for there is no hope of attaining the 

 imaginary forms accurately, when we employ approximations 

 only to the real quantities which enter into the expression of 

 them. 



Although, as stated at the outset, it is not my intention at 

 present to enter into any discussion of the forms which follow 

 (A), yet I may perhaps be permitted briefly to notice here 

 one or two obvious deductions from them. 



By putting s for <^{x), we at once see how easily the usual 

 formulas for the solution of equations of the fourth degree may 

 be obtained from those forms : we shall only have to multiply 

 together the quadratic factors 



x'^-2Yx+fs 



and to equate the resulting coefficients with those of the like 

 powers of .r in the proposed equation : and it may not be un- 

 deserving of notice, that when two roots are reciprocals, and 

 two only, then/=l, and F, F' may each be determined by a 

 simple equation. 



