76 



Proceedings of the Royal Irish Academy. 



Note added in Press. — The following procedure enables us to 

 avoid incommensurable subjects. In the selected form of the original 

 equation separate the term free of x into two terms j + ; j is now 

 made the coefficient of on the left of the equation, and 'J^", that is, 

 k, is made the new subject. For example, x^ - 2x - b = Q may be 

 written - 2^ + 3 = 8 ; and ^8 is commensurable. The same roots 

 are given by aid of the Table, but care must be taken not to retard 

 the convergency by this process, which, however, helps us in other 

 ways. 



From the solution of many numerical equations it appears that a 

 real subject always gives the greatest or the least root, or both. 

 Two more may often be obtained from the equation in s. The evalua- 

 tion of inverts with unreal subjects cannot be discussed in these very 

 brief notes on a large theme. 



