Eoss — Verh-Functions. 



33 



But operations must have operative as well as algebraic powers of 

 combination. Tor instance, when a + b(3 is multiplied into x, the 

 result is 



(a + hp) X = ax'+ bxjS ; 



but when it operates on the subject x, the result is -f bx. We must 

 therefore be careful always to distinguish between operative and 

 algebraic relations. Thus, as we are now able to render explicitly 

 any operation it will no longer suffice to write without defining 

 whether we mean algebraic or operative involution. We must, in 

 short, employ a special bracket for operative relations ; and the square 

 bracket is the most convenient. Hence [_a+ b(i~\ x denotes that a+b^ 

 operates on x, and [<^]" denotes that </> operates on itself n - 1 times. 

 On the other hand, {a + b/3) x and (<^)" denote algebraic relations. 

 The following will serve as first examples : — 



c+~'j [e^]0 = a-b{c + d)- 



[a - b(3]ip-^-] 



[«^^/3r =^^, [log(«+^)]- 



[a + bfif =a + ab + ab' ab*'-' + b''/3, 



a -\- a + a + ' ' ' a + b(3 ^ 



It will, of course, be understood that the subject of an operation 

 should always be placed after it. The subject of an operation need 

 not be in square brackets, unless it operates on another subject ; and 

 the square brackets may often be omitted for recognized operative 

 symbols such as c^, xj/, A, D, when it is clearly understood that 

 only operative relations are being discussed at the moment. A 

 single stop between two symbols may be taken always to imply 

 multiplication, as in <}!> . xf/^ and a double stop, as in ^ : i/^, to mean the 

 same thing as the square brackets. 



4. We have defined /3 as the symbol which denotes the base of an 

 operation, that is, the position which the subject will occupy when the 

 operation is performed. But it may be otherwise interpreted. Accord- 

 ing to our conventions the function x" becomes \^(3"]x when put in the 

 form of operation and subject. Hence jS'* is an operation which raises its 



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