182 Mr. Babbage's essay towards 
we have a very convenient mode of doing it ; these operations 
would be thus expressed ; 
2 j 3> I > 4> 3 
, G 2 > 3> 4j 3> 2 
This notation may not appear sufficiently concise to those who 
do not consider the very complicated relation expressed by 
the above written symbol : it need, however, only be used 
in very few cases, and when the lower series of indices is 
omitted, it must always be understood, that the quantities 
themselves are arranged in the order in which they are to be 
operated on. 
If in a function of two variable $ [x,y,) we take the se- 
cond function relative to a;, and then the second function rela- 
tive to y, we have 
2,2 
If we take the second function first relative to y , and then 
the second relative to x we shall find 
2 , 2 
4 2 ’ \x,y)=^ { 4> W [x>y)>y ) } 
It appears from this, that the order in which these operations 
are performed is not immaterial, as the order in which we dif- 
ferentiate a function of two variables, is in the differential cal- 
culus. 
The two expressions just given are the two second func- 
tions of t|/ {x,y)> the first taken relative to x and y, and the 
second taken relative toy and x > But there may be another 
second function of f which will arise from substituting 
at the same time $ (x t y) for x, and ^(x,y) for y, it will be 
