780 PEOFESSOE BOOLE OX THE CO^NIPAEISOX OE TEAXSCEXHEXTS, VITH 
become infinite for the critical values Xj, Xg-.X^, we mjy^ose x to app'oach each enticed 
value by equivalent infinitesimal variations. 
The most important, perhaps the only important, cases are those in which f{v) vanishes 
when V is infinite. 
33. I shall begin with noticing some particular deductions from the theorem, among 
which will be included certain known formulse of analysis. I shall show that it enables 
us to deduce from any known definite integral the values of an infinite number of other 
definite integrals of progressively increasing complexity. I shall show that when the 
arbitrary function under the sign of integration is regarded as discontinuous, the first 
member of the equation becomes resolved into a number of definite integrals of con- 
tinuous functions, and that we thus arrive at the same theorem for the comparison of 
functional transcendents, (5.), art. 30, from which the above theorem was itself derived. 
Finally, I shall apply the theorem to the extension of the theor}' of definite multiple 
integrals. 
Special deductions may be obtained by limiting either, — 1st, the form of the rational 
function <p{x) ; or 2ndly, the interpretation of the functional sign/*; or 3rdly, the number 
and value of the constants in the function under the sign f. 
1st. Let (p(x)=l. Then in the second member of (1.), art. 32, 
e[?W] 
v—x-\- 
dv 
j dn 
X — \ X — Xn 
Cl 
dv 
V — X + 
a j 
a;— Aj x—X, 
Hence 
dv 
x—v 
x—X^ 
=dv. 
dn 
r/(*- 
(2) 
This was the theorem, or rather the most important case of the theorem referred to 
in Art. 1, as published in the ‘ Cambridge and Dublin Mathematical Joiunal,’ Xo. XIX. 
The following are special applications of it, chiefly selected from the paper in question. 
Since we have 

and 
we shall have 
J_ f{v’'-^2a)dv (4.) 
Hence 
P (5.) 
