INDEX OF A FUNCTION. 277 
f@= Jere 
f° @)=2% 
Sf? @=Ve+V/ate 
Oe lela 
fori? fa (@)=Vat+e? a= Vti=g 
fr @=er—)*—s 
woates 
ea oes mat , 
to n brackets. 
Note :—Since writing the above, the invaluable treatise of Pro- 
fessor Boole on Differential Equations has been published. In his 
XVIth chapter there are a few remarks on inverse forms, which seem 
to bear out what has been said on their proper interpretation. He 
writes, commenting on the index laws as applied to functions: “ All 
that is said above relates to the performance of operations definite 
in character upon subjects proposed to be given. But an inverse 
problem is suggested in which it is required to determine, not what 
will be the result of performing a certain operation upon a given 
subject, but upon what subject a certain operation must be performed 
in order to lead to a given result.” So below he adds: “If w repre- 
sent any operation or series of operations possible when their sub- 
ject is given, and then termed direct, and if in the equation 7 u = v 
the subject ~ be not given but only the result = » then we may 
write «= 214, And the problem or enguiry contained in the 
inverse notation will be answered when we have, by whatever pro- 
cess, so determined the function w as to sstisfy ru = vor7a 149 
=v. By the latter equation the inverse symbol z~— ! is defined. 
Thus it is the office of the inverse symbol to > Propose a question, not 
to describe an operation.” 
If the inverse symbol has an office, it is obviously more than a 
mere convenient notation. The form of the above statement may 
perhaps be open to objection, since when two precisely reverse opera~ 
tions are performed it seems as fair to denote one of them a question 
as the other. But the view taken of the inverse symbol is the same, 
whatever be thought of the propriety of this statement. 
