38 Proceedings of the Royal Irish Academy. 



Let now each of the roots of the equations [ 1 ] and [2] be diminished 

 by any positive number 2 ; then the signs of the terras in the trans- 

 formed equation deduced from [IJ will, all be positive; but in the 

 transformed equation deduced from [2] it is plain that h may be suf- 

 ficiently small to cause the signs of the terms to be alternately positive 

 and negative, as in the equation [2] itself. The same pair of ima- 

 ginary roots must, however, enter into loth of the transformed equa- 

 tions; thougb, in the first of these, the signs of the terms imply that 

 the imaginary roots belong to the negative region ; and in the second, 

 the signs imply that these same roots belong to the positive region. 



"We thus see that the region in which a pair of imaginary roots is 

 indicated by the signs of the terms of an equation, supplies no suffi- 

 cient clue as to whether the real part, common to the two imaginary 

 roots, is positive or negative, whenever the equation is of a higher de- 

 gree than the second degree. In a quadratic equation, of which the 

 roots are imaginary, the case is different ; the sign of the real part ia 

 always indicated ; it is the opposite of that prefixed to the coefficient 

 of the middle term, whenever such middle term is present. If the 

 middle term be absent, the real part of each imaginary root will then, 

 of course, be zero ; but in a complete quadratic equation, not only is 

 the region to which each of the two roots belongs indicated — whether 

 these roots be real or imaginary — but the part of the pair of roots which 

 is free from the radical sign is also indicated ; it is always half the co- 

 efficient of the middle term, taken with changed sign — the coefficient 

 of the first term being unity. In other words, as before stated^ the part 

 which precedes the radical sign is always the root of the simple equa- 

 tion derived from the quadratic, whether the other part of the expres- 

 sion for the pair of roots of the quadratic be real or imaginary. 



In the foregoing discussion, I have frequently spoken of a pair of 

 imaginary roots as being in, or belonging to, the positive region, or 

 the negative region. This phraseology, though in conformity with 

 general usage, is objectionable. An imaginary quantity cannot have 

 any place in a series of positive numbers, nor yet in a series of nega- 

 tive numbers ; because it is entirely out of the range of every series of 

 numbers. The phrase should be taken to mean merely that the indicator 

 of the pair is in the positive, or in the negative region, as the case may 

 be ; or, still more explicitly, that the indicator is a positive or a nega- 

 tive number. We have seen above that the same pair of imaginary 

 roots may have just as much claim to a place in a series of negative 

 numbers, as to a place in a series of positive numbers, and to a place 

 in a series of positive numbers, as to a place in a series of negative 

 numbers ; and this is the same as saying that the pair itself is not en- 

 titled to a place in either series : it is the indicator only of that pair, 

 which can be said, in strictness, to range with positive or with negative 

 numbers, or which can occupy any place among them. 



Taking the derived functions in reverse order, commencing with 

 the function of the first degree, if the value (?-i) of .c, which causes 



