the calculus of functions. 181 
The first index 2 refers to x, and the second index 1 refers 
to y. Similarly if instead of y in the original function, the 
function itself had been substituted, the result would have been 
the second function relative to y; it would be thus denoted 
(-M' (^y)) 
If there are more than two variables in the original function, 
they may be arranged in the order in which they are to be ope- 
rated on, and the indices will denote the number of operations 
to be performed. 
Thus ^ 2,3,r ’ 4 (.r, y, z, v,) signifies that in the function 
■ty {x,y, z, v,) we must instead of x substitute the function 
itself, and in the result instead of y put the same function, 
this latter operation must be repeated, and finally, instead of 
v in the last result, put the original function ; this last opera- 
tion must again be repeated twice. 
There are many cases which this notation does not com- 
prehend. If, for example, in the function just proposed, we 
wished again to take the function relative to x o ry, it would 
not be easy to express this. The method I propose is to have 
two ranks of indices, the lower one to distinguish the quanti- 
ties operated on ; the upper one to mark the number of ope- 
rations performed. According to this method the example 
just chosen would be written thus ; 
2, 3, n 4 
. I > 2 > 3 > 4 
r(x f y,z,v) 
If only such functions as these occur, we encumber our sym- 
bol without any advantage ; if, however, we now wish to per- 
form any farther operations, such, for instance, as to take the 
second function relative to z, and then the third relative to y, 
B b 2 
