2]
SYMBOLICAL EXPRESSION.
II
Or the function <f>(a — x, b — y...) not becoming infinite within the limits of the
integration, we have
ff... (f) (a — x, b — y ...)dxdy ...
‘‘¿Ilk, ... 7T*”
£
r (|n) ^°2^ +1 .1.2..._p4n(|n + lj...(Jn+jD)r da-
the integral on the first side of the equation extending to all real values of x, y„
h^h-
1
0C~ f lï~‘
&c., subject to -t^ + To+ ••• < 1.
Suppose in the first place <6 (a, b ..0 = -
F V (a 2 + b 2 ...f n
By a preceding formula the second side of the equation reduces itself to
2hh,... 7r in f 1 a?” -1 dx
y being given by
Hence the formula
r (in) J o V{{rf + h 2 x 2 ) {if + h 2 x 2 ) ... (n factors)} ’
a 2 6 2
if + h 2 rf + hf ’ “
dxdy...
... n times
2hh,... 7r in f 1
{(a-tf) 2 +(£-y) 2 ... } in
¿c n 1 . dx
r (in) Jo VK 7 ?' 2 + fdx-) {rf + hfx 2 ) ... (n factors)} ’
where the integral on the first side of the equation extends to all real values of
x, y, &c. satisfying j- +y- + &c. ... < 1; if, as we have seen, is determined by
a 2 b- , 0
^+^ + ^+^ 2+&C - -1;
and finally, the condition of <p{a — x, b—y...) not becoming infinite within the limits
a- b'
~hr- + h:
of the integration, reduces itself to ~ 4-... > 1, which must be satisfied by these
quantities.
Suppose in the next place that the function </> (a, b...) satisfies + &c. = 0.
f d 2 \
The factor f/¿ 2 &c.J may be written under the form
+<*«■ ~ ^ +&c - + » (i + m ■■)=<*• - ,i!) w +&c -
2—2