787] FUNCTION. 
535 
or as this may also be written, neglecting the negative powers of x, 
Tlx = V27r exp {(¿c + log x — x). 
Another formula is IVT (1 — x) = —: or, as this may also be written, 
sm 7tx J 
II (x — 1) II (— x) = — 
sm irx 
It is to be observed that the function II serves to express the product of a set 
of factors in arithmetical progression; we have 
(x + a) (x + 2a) ... (x + ma) = a m ^ + 1^ ^ + 2^ ... ^ + nij = a m Yl + m)j -=- II ^. 
We can consequently express by means of it the product of any number of the factors 
which present themselves in the factorial expression of sin a. Starting from the form 
uUr (l + —) iWl-—), 
V sir J \ sir J 
where II is here as before the sign of a product of factors corresponding to the 
different integer values of s, this is thus converted into 
uU + m) n ( -1 + n) - n ^ n {-1) nmTin, 
or as this may also be written, 
ttII (- + m) n + w) n - l)n (- ^ Tim Tin, 
which, in virtue of 
becomes 
n lln(--l = -7 
it sm u 
= sin uTl + m J II ^ + n 
TlmTln. 
Here m and n are large and positive; calculating the second factor by means of the 
formula for Tlx, in this case we have the before-mentioned formula 
uTl 1 m [l + —)lWl- —) = (-) sin u. 
' sir \ sir \m> 
The gamma or n function is the so-called second Eulerian integral; the first Eulerian 
integral 
f xP~ l (1 — x)9 -1 dx, = -4- r (p + q), 
J 0 
is at once expressible in terms of T, and is therefore not a new function to be con 
sidered.