222 SCIENCE PROGRESS 



used in order to distinguish it from an algebraic ratio. Thus 



if <f> = x^> = — X- As however operations in operative multi- 



plication are not commutative as are numbers in numeral 

 multiplication — that is, <f>yfr J does not always = y}r<f> — so we may 

 have to distinguish between pf -1 and ^ -1 $ ; and the former 

 may therefore be also written (f>^yfr and the latter ^\^> — but 

 this is seldom required. The following propositions are almost 

 self-evident : 



<£ (jj -\- yjr <f) yfr 0-v/r <f) <£ ty (f) 



5 x x x x¥ x <rx x 



and we must remember that [^] _1 = yfr' 1 ^' 1 . Obviously also 



9 

 But operative ratio may exist between numbers as well as 



y 



between operations. If y=(f>x, we may write = = <f> ; so 



that this operative ratio denotes the operation which connects 

 the two quantities y and x. But here we tacitly assume that 

 y and x are variable quantities, capable of an infinite number 

 of values, all connected by the operation 9. On the other 



hand the ratio =, where a and b are single quantities, will 



denote an infinite number of operations capable of connecting 



them. In other words, the operative ratio f denotes any 



b 



curve which passes through the point of which the co-ordinates are 



a and b. If the same curve also passes through the points 



(c, d), (e, /), etc., we have the series of equations 



and if the form of <f> is given, we can determine its constants 

 by the Euler-Lagrange interpolation formula or otherwise. 

 Where the form of 9 is not given, it can be determined if the 

 number of equations, that is, of points passed through, is large 

 enough. 



1 The square brackets may be neglected where the meaning is obvious without 

 them. For the numerical product of two operations such as <£ and }fr can be 

 clearly expressed by <p . ■*//■, and the numerical powers of <\> by ($)". 



