44 



Proceedings of the Royal Irish Academy. 



Of course, only particular expressions can be put into operative 

 factors in this way; because, as noted in § 11, the coefficients of an 

 operative product are not as a rule independent. But if the coefficients 

 are numerical, their interdependence cannot always be detected at sight, 

 and operative division then affords a quick and easily remembered 

 method for testing the point, and at the same time attaining the 

 object, if such be possible. 



It is sometimes possible to jyut an operation in the form of an 

 operative ratio. Then, if 



= J, we have = ^ ' 



For this purpose, we have to find both the divisor and the dividend — 

 the quotient being known. Take, for instance, the cubic 



x-" + ax 



and assume the forms 



P+pI^-\ and (S' + q/3-' 



for the divisor and dividend which have the quotient + afS. 

 we find that 



3p = - a, and 9q =^ - a^, 



m that 



Then 



13' + 



^-(31/5- 



which is Cardan's solution in substance (§15-7). This section would 

 be capable of much more development ; but it is advisable in a preli- 

 minary survey to proceed quickly to the next. 



15. General Solution of JEquations hy Operative Division. — Since 



[<;.]-' [<^] = p, [0]-' 



and if <^ be a linear function of /3, we can in general develop = in a 



9 



series bv means of operative division, just as we can develop — in a 



series by algebraic division. The quotient will be in ascending or 

 descending powers of yS, according to whether the divisor is arranged 

 in ascending or descending powers ; and the process is the same as 



