Ross — Verb' Functions. 



39 



by the whole of j/^ ; and we may then rearrange the terms in ascending 

 powers of /?. For example, 



Both sides of the equation may now be applied to any subject, quanti- 

 tative or operative. 



To obtain the general result, the particular results produced by 

 each term of the superior factor should be written out below each 

 other in columns of the same power of yS, and the sum placed at the 

 bottom. If any of the exponents of /? in the superior factor are 

 negative, the result will contain fractions which can be dealt with in 

 the usual algebraic manner. If any of these exponents are fractional, 

 we can reduce them all to a common denominator, as in the following 

 example : — 



[i?/3^ + + r/?^] ^\f = [pfi'''^+ ql^'^^ + r'''] IP^'^] if/ 



and if/ = (»/^)^. 



If P" and jS"" be the highest powers of jS in the two factors, then 

 will be the highest power of /3 in the result, which will, in the 

 general form, contain 9im + 1 terms. It may be observed that, as 

 we shall, in general, have 7im + 1 equations to determine the coef- 

 ficients of these terms, and as the original factors can contain only 

 n + m + 2 independent coefficients, the coefficients of the result will 

 not generally be independent. 



"WTiere the two factors are the same the result is an operative 

 square. It may be noted that [+ (S^'J = + (3''" ; that [- jS^'J = - ^8"' if 

 n be an even integer, and that 



Obviously the rule of operative multiplication differs from that of 

 algebraic multiplication only in the fact that every term of the 

 superior factor operates on the whole of the inferior factor, instead of 

 being multiplied into it. 



12. Operative Fractions. — In extension of the phraseology em- 

 ployed above, [(^] [i/^]"^ may be called an operative fraction, and may 



also be written = — a double line being used to distinguish it from an 



