Eoss — Verb-Functions. 



35 



5. We must now examine the operative potency of a simple 

 quantity. Consider the linear operation 



a + + c/r^ . . , 

 When this operates on the result is, by definition, 



a + + cx^ . . . 



We may write this as follows : — 



la+h/B + . . = ]^a']x + [h/3']x +;[^y8-]^ + . . . 

 = a + hx + cx'^ . . . 



Thus, while has squared the subject, and has simply repro- 

 duced it, \_a] has reduced it to unity. To explain this, we observe 

 that, as is merely the algebraic power of p, it follows that, accord- 

 ing to algebraic rule, = 1. Hence the original operation may be 

 written 



-^h^ + cp' . . 



so that 



= [_ap^^x = ax° = a. 



If 



= x'\ then obviously = x'^ = I. 



Hence a ''free" quantity when in operation merely reproduces itself. 

 Quite rightly it appears in the result, because it is not zero; but 

 equally rightly it has no effect on the subject, just because it is a 

 quantity and not an operation. For, consider if it is to have an effect 

 on the subject, what effect is it to have ? If \_a^x does not equal a, 

 does it equal a x, or ax, or a"", or log^^ ? It cannot equal any of 

 these, because they are respectively the results of 



[a + fS^x, [a/S^x, [a^']x, and [\ogaP']x ; 



and there is, a priori, no reason why it should equal any one of them 

 to the exclusion of the others. 



6. Lastly, we have to show that [</)]°, or, as it is commonly 

 written, <^°, cannot be equal to unity. For 



[<i>y^ = [<f>rwx = x', 



but, if 



W=l, then ll']x = x. 



But we have just seen that [1]^ = 1 ; and the two results are not 

 compatible. 



