234 
2, Similarly, if we were required to determine the value of a definite 
integral of the form, 
‘es F (v-1) #dz, 
Pat 
the value of a being, as before, independent of the limits of the integral, 
we have, as our symbolic result, in the first instance, 
P0)-(ST) 
(the suffix to the symbol A being employed to denote that this symbol 
is understood to operate only upon a), and the further evaluation, as be- 
fore, simply depends upon the particular form of the given function F. 
3. More generally, if we were required to determine the value of any 
definite integral of the form 
% (¥Y2 a P 
[2 P @ EW) a ytdedy, 
wherein the quantities a, 8, are supposed to be independent of the limits" 
and of each other, we see that this integral is equivalent to 
F, (ea) F, a - ie ( xryPda dy, 
or 
x2 ne (428! — y,F) dx: 
F, (¢%2) F, (272) - B+1 +1 als, 
and if, for simplicity, the equations of the limiting curves be written in 
the form 
y.P" =f (2); yi" =f, (a), 
the final symbolical value of the given integral is 
B, (eo) Bae?) «go (fa ea) fe (0Pe)}. = a 
a+l1 
and the complete evaluation now depends, in general, solely on the par- 
ticular values of the given functions F,, F., f,, f. 
4, Similarly, if it were proposed to determine the value of a definite 
integral of the form 
belie (2-1) F,(y-1) «2 y@dady, 
Eat 
wherein the quantities a, 8, are supposed to be independent of the limits 
and of each other, we see that this integral is equivalent to 
F, (Aa) -F2 (Ag) ne is arybdedy, 
or, with the same Bist curves as before, 
Fs (Aa) Fa(Ap) gq tele) ~A (e?)} (AE), 
