42 Proceedings of the Royal Irish Academy. 



The reason for this is, that the divisor itself is the operative product 

 of 5 + 2)8, and /? + and 



'Ax X 



It will be seen that the quotient obtained by ascending division easily 

 reduces in this case to that obtained by descending division ; but it 

 would have been simpler, before undertaking the ascending division, 

 to have put the divisor into factors, and then to have divided /? by 



/J + 



This example would serve to solve the functional equation 

 f{x- + 2^+5) = + 4^3 ^ 17^2 ^ 26^ + 44 ; 



but the solution can be otherwise obtained, though not so quickly; 

 and the example is given only as an easy illustration of the general 

 processes of descending and ascending operative division. 



14. Separation into Factors. — In the above instance the divisor 

 was known ; but a more important case occurs when both divisor and 

 quotient have to be determined — when, in fact, we are required to 

 put a linear operation into operative factors. Por, if [^] x = y is an 

 equation which we have to solve ; if we can find two factors, i/^ and x, 

 such that = and can readily find the values of \^~^ and ; then, 

 since <^~^ = x~^'A~S ^^n solve the original equation. Probably, the 

 readiest way to achieve this is to assume the form of the divisor, and 

 then ascertain by division whether it will produce a suitable quotient. 



For example, solve the equation 



147;^;^ + 42^3 + 10^2 + :r = 30. 

 Then we have to put 



yt?+ 10)82 + 42^3^ 147)8* 



into factors. 



Try division by /8 + a^^. 



P + ajS^J /3 + 10)82 + 42)83 + 147^* 1/3 + (10 - a) 13^ 

 /3 + ap^ 



(10 -«))82 + 42/33+ 147)8* 



(10 - )82 + 2« (10 - a) + «2 (10 - a) )8* 



