since [iS"][/S™] = /5'"» and = 



40 Proceedings of the Royal Irish Academy. 



algebraic fraction. It must be noted that [i/^]"^ [<^] is not the same as 



= as just defined. The former may be called a superior operative 



ratio, the latter an inferior one ; and they may also be rendered by 



xlf \ <^ and 4> // xp. 



The elementary properties of these fractions are almost self- 

 evident ; for example — 



t = a, ^t^t^, but 



^ XX XX 



13. Operative Division. — If the denominator be an operation which 

 consists of two or more terms, we can often find the quotient by 

 processes similar to those used in algebraic division. Suppose, for 

 instance, that 



[<^] + 2/3 + o'] = (3' + 4(3' + 17^2 + 26/3 + 44, 



and that the value of <f> is required. We may proceed either by 

 synthetic or formal division. If the former be selected, assume that 



<}> = (3- a(3 + h, 



and operate with this on 



(3^ + 2/3 + 5, 



and then equate the coefficients of the various powers of (3 in the new 

 equation. We find that 



a+14: = 17, 2a + 20 = 26, 5a + b + 25 = 44, 

 so that a = 3 and b = 4. 



Or we may proceed as follows by means of a formal rule of 

 operative division closely similar to that of algebraic division: 



(3^ + 2/3 + 5J /3' + 4ft' + 17/3^ + 26/3 + 44 1/3' + 3(3 + 4 



(3' + 4(3^ + 14/52 + 20yS + 25 



3/32 +6(3+19 

 3(3'+ 6(3+ 15 



4 

 4 



