744 
PBOEESSOE CAYLEY ON PEEPOTENTIALS. 
expressions, the elements are finite, and the corresponding portion of the integral is 
indefinitely small. We may consequently reduce as above ; viz. writing now 
R=l- 
the formula is 
/2 -. A „ 
<?<*' • 2m(E -9*-'£t/ . . . h)~\ 
= — 2m(/. ..h)-' . f « . e(E-|)— ; 
Jo 
citing the integral becomes =R, ?+m f duu q (l—u) m ~\ which is 
r ( i + g ) r ( m ) m m 
' r(l + g , + m ) ’ 
that is, we have 
and consequently 
that is 
^d_w = _ 2(f h) . K 
r ( i + g ) r ( i + w ) 
r(l + g + m ) ’ 
r(2 s +?) or -p 7A-i f(i +5 , )r(i + to ) 
2(ri)T(l + g) J * T(\+q + m) ’ 
— ( -f 7A-I r(¥ s +g)f(l + m ) T?9+>» 
? 1/ • • • n ) (ri^r^+g+m) 5 
viz. g has this value forvalues of ( x...z ) such that butis=0 if ^...-f^>l. 
J h J h 
110. Multiplying by a constant factor so as to reduce g to the value R ?+m , the final 
result is 
q+m 
•~Tf) 
■_c (*-£•• 
J [(a— ») 2 ...H 
. ~\2 I 
the limits being given by the equation 
~2 
- .. + -=1 
is 
if - rfp -'-' - ~ 7uP t + f ’- ■ ■ t+h ^’ 
where 6 is the positive root of 
0+/2 e+^2 fl 
eo-f.. 
z\ ?+>» 
-p) *•••* 
J {{a—xf. 
In particular if e=0, or 
