AND ON DEFINITE INTEGRATION. 481 
the late Mr. Murphy’s admirable Treatise on the 
Theory of Algebraical Equations, page 101; and 
the Differential and Integral Calculus, by Pro- 
fessor De Morgan, page 311], that each of these 
Mathematicians had investigated a somewhat si- 
milar theorem, given originally by Euler, but in 
a form less practicable than the above, and de- 
rived from entirely different principles. Nor do 
these very eminent Mathematicians appear ever 
to have thought of applying their theorem to the 
approximation of definite integrals. 
It now remains for me to apply the above ge- 
neral formula, by way of illustration, to a few 
examples. 
Ex. (1.)—Required to approximate to the 
integral of log. v dx, between the limits y=11 
and #=20. 
I 
Here, we have f (x) = log. a «. f (v7) = =. ; 
Ifl Vv 
ae (2 Vs ee ik G3 ae ei, Gc, 
And, if we substitute these values in formula 
(15), we shall have, by restoring the values of 
A, B, D, &c., &c., 
