353 



words, if r be the transforming factor by which the second term is re- 

 moved, the imaginary pair, traceable to the pair in the cubic (7 = 0, is 

 indicated between r — § and r + $ : since it is the leading triad of E that 

 fulfils the condition- — the same condition, by (7), as that fulfilled, by 

 the leading triad of C. No doubt, in certain cases, other coefficients, 

 besides the second coefficient, may also vanish between like signs for 

 the same transformation (r), and other pairs of imaginary roots be in- 

 dicated between r - & and r + <5. But, by taking account of only a 

 single pair, in all circumstances — the pair, namely, that would neces- 

 sarily be an imaginary pair, though any or all of the coefficients in B, 

 after the leading triad, were changed — we restrict ourselves, as we 

 ought, to that pair alone, the imaginarity of which is conveyed, in the 

 reverse process, through all the intermediate equations, from (7 = 0, up 

 to E = 0, regardless of whatever other imaginary pairs may be, as it 

 were, picked up and absorbed into the equation in its progress towards 

 completion from C to B. Whatever modifications this pair may under- 

 go from changes in the coefficients after the third term, its character as 

 an imaginary pair in B = 0 is still preserved, and it continues through- 

 out to be indicated between r - B and r + 8. 



It will have been observed, that in the foregoing reasoning B, is re- 

 garded as derived from C (after restoring at each step the factor pre- 

 viously expunged) by the process of integration, as we may for the 

 moment call it : and it is to be noticed that it is the general integral, at 

 each step that owes a pair of imaginary roots to the entrance of 

 the pair in C = 0 ; in other words, that although the constant 

 completing any integral be taken of any arbitrary value, even zero, a 

 pair of imaginary roots — primary roots, resulting from the pair in 

 (7=0, must still enter. The constants actually introduced are each of 

 assigned value, because a specific equation, E = 0, with assigned coeffi- 

 cients, is ultimately to be deduced. The constants, added one after 

 another, as the derivation (or integration) proceeds, may cause the in- 

 troduction of additional imaginary pairs, as just noticed ; but none of 

 these pairs are traceable to the pair in (7=0; and a pair traceable to the 

 pair in C = 0 would still enter each of the ascending equations, though 

 no constants at all were introduced. 



(17) From what has now been shown, we see — always bearing the 

 general property (7) in mind — that the search after distinct and inde- 

 pendent primary pairs in the equation [/] may be converted into a 

 search after the independent primary pairs in the group of cubic 

 equations [77] ; for although, in applying the tests [3] to each of these 

 cubics, n is equal to 3, yet that when the blanks in the coefficients 

 are filled up, the conditions, unaffected by this lower value of the ex- 

 ponent, become the very same as those marked [3], in which n is the 

 leading exponent of the equation from which these cubics have been 

 derived. So far, therefore, as this inquiry is concerned, the group [77] 

 effectually replaces the single equation [J], with this advantage, namely, 

 that as the individual equations [77] are connected together so that the 

 final triad of coefficients of one supplies the leading triad of coefficients 



