18 
ON CERTAIN DEFINITE INTEGRALS. 
[3 
j dr u ) 
Substituting this value, also / + + g-th 2+ •' • j ^ or i n va -l ue of U, 
and observing that m — 0 gives % = oo, m = 1 gives % = £, where £ is a quantity 
determined as before by the equation 
a 2 b 2 t _ 1 
F+A s + f + V + "' _ ’ 
we have 
tó, ..Va ft ^{f + V "\ d ? 
r(in) J„(f+A’W{(f+TT<£ +A,*)•••]’ 
or writing (f> + £ for di;' = d(f>, the limits of 0 are 0, » ; or, inverting the limits and 
omitting the negative sign, 
-M, .„-■**» r •Of + fr l + <i + g4/i, s + 0 + -"| ri * 
T (^n) Jo (£ + /¿ 2 + <£) V[(£ + + 0) (£ + h 2 + (f>)...{ ’ 
which, in the particular case of n = 3, may easily be made to coincide with known 
results. The analogous integral 
JJ... (n times) 
f% + h + ' t \ dxdy ”‘ 
{(a-x) 2 +(b- y y...r 
is apparently not reducible to a single integral.