i8o Mr. Babbage's essay towards 
regulate the action of the ultimate particles of bodies. To the 
accomplishment of these desirable purposes, it must be con- 
fessed that it is in its present state unequal ; but should the 
labours of future enquirers give to it that perfection, which 
other methods of investigation have attained, it is not too 
much to hope, that its maturer age shall unveil the hidden laws 
which govern the phenomena of magnetic, electric, or even 
of chemical action. 
When functional equations containing two or more variables 
occur, their solution presents still greater difficulties than 
those we have already considered ; the new relations which 
arise, necessarily require a new notation to distinguish them. 
I shall endeavour, as far as I am able, to apply or extend 
that alread}^ in use; but, as it is almost impossible in the 
infancy of a calculus to foresee the extent to which it may be 
carried, or the new views which it may be necessary to take 
of it, the notation I have used should only be considered as 
of a temporary nature ; it may be employed until some more 
convenient one be devised: perhaps, however, it might be 
more advantageous that it should not be altered until our 
acquaintance with this subject becomes more intimate, and 
until the infinitely varied and comprehensive relations dis- 
played in the doctrine of functions, have been more minutely 
examined. 
If 4/ be the characteristic of any function of two quantities 
x and y, that function is thus denoted ^ {x, y). Now, if in- 
stead of x in this quantity the original function be substituted, 
I shall call the result the second function relative to x, and I 
shall denote it thus 
= 4 / {■b(x,y),y) 
