2,5^ Mr. Babbage’s essay, &c. 
To complete the outline of this new method of calculation, 
it would be necessary to treat of equations involving two or 
more functional characteristics, and to explain methods of eli- 
minating all but one of them : these lead to a variety of inte- 
resting and difficult enquiries, and will probably be of consider- 
able use in completing the solutions of partial differential equa- 
tions : it would also be proper to consider the maxima and 
minima of functions, and to apply to this subject the method 
of variations ; these are points of considerable difficulty, and 
although I have made some little progress in each of them, I 
shall forbear for the present any farther discussion on this 
subject. In the mean time, the sketch which I have offered, 
and the few applications I have given, are sufficient to point 
out the great importance of this method. It should however 
be observed, that its applications have only been noticed inci- 
dentally ; my object has been to direct the attention of the 
analyst to a new branch of the science, and to point out the 
manner of treating it : the doctrine of functions is of so general 
a nature, that it is applicable to every part of mathematical 
enquiry, and seems eminently qualified to reduce into one 
regular and uniform system the diversified methods and 
scattered artifices of the modern analysis ; from its compre- 
hensive nature, it is fitted for the systematic arrangement of 
the science, and from the new and singular relations which 
it expresses, it is admirably adapted for farther improvements 
and discoveries. 
