Mr. Booxe on the Analysis of Discontinuous Functions. 125 
so clear and explicit, but that, in point of fact, the views which are very com- 
monly held on this subject are vague and unsatisfactory, as recent discussions 
have tended to show. A hope is entertained that the considerations by which the 
theorem is here deduced will serve to render its real nature less doubtful. In a 
supplementary chapter we shall, by way of further illustration, adduce some 
examples of integrals, which are, in like manner, to be considered as the limits 
to which more general integrals approach, as two quantities, & and k’, approach 
to 0. This will throw some light on the cause of the difference which is some- 
times observed between series and their envelopments, when submitted to a pro- 
cess of definite integration, and will also explain the appearance of the imaginary 
factor in integrals, the subjects of which become infinite within the limits of inte- 
gration. We shall conclude this part of the subject by an application, which, 
though it has no special bearing on the principles above stated, may be thought 
to possess an independent interest. By the aid of Fourier’s theorem, from the 
equation f(w) = « will be deduced a symbolical expression for the value of F(w), 
involving an arbitrary function of x, independent of the form of either f(w) or 
F(w). Particular determinations of this function will lead us to the known 
theorems of Lagrange and Laplace, while its arbitrary character will show that 
these belong to a class of theorems of which the number is infinite. 
1. Those who, commencing the study of the differential calculus at a mature 
age, bring to it amind disciplined by the pursuit of other sciences, feel an insuper- 
able difficulty in admitting, what some writers assert as an axiom or first principle. 
that what is true up to the limit is true at the limit. And doubtless this is an 
unsound induction. The only condition under which we are permitted to pass 
from a series of particular truths to some general proposition is, that every truth 
included in the general may be proved in the particular; unless, indeed, we assert 
an a@ priori principle of continuity, antecedent to all experience, and independent 
of all proof, which were to abandon the dominion of reason for that of intuition. 
But not only is the axiom alluded to unfounded,—it is unnecessary to the purpose 
which it is thought to subserve. The differential calculus is the calculus of 
limits, not of values actually realized. What is, quantitatively, the value of a 
