PROFESSOR CAYLEY ON PREPOTENTIALS, 
where the rT-functions on the right hand are respectively 
_ fi qfr+g-ifi-afl-g-frfe 
Jo (x 2 -/ 2 tf) 9+ * 
_ +2g r ^ g -*( \-x)*-i-'dx 
Jo (x 2 -/ 2 «)^ + « 
“* Jo 
~ Z Jo (« 2 -/^)- g+i 
we have then 
723 
=-^=jy, y _r*+’{t+F-*?F*'-'(t+F)-<-* at, 
j,s+2q /»“ 
=v=jt’ £_/"* v+f-**)’- 1 (t+fyi-’dt, 
= (x‘- r . _J r ^ > 
V-2J+1 F“ 
=S?r^J M ; 
fs K i-s TUoTa F" 
Eing-integral=-j^p; r( ^ + = )r( \_ g) j ^J-^t+f-xJ-^t+f)-’-* at 
fs ruora F“ 
= r (i + q) r (Is - q ) J K2 _/ ?_i (^+/ 2 -^)’“ 21 {t+f)~ is - q dt 
= /* r ft4)r(i +? ) £ / r g ~ l v+r-*r'- i v+fy^at 
fsy\-s r+sF 1 F” 
= W^ c T l r (^- ?) r (i + ?) J K2 _ /2 r is_? ^ at. 
Observe that in II. and III. the integrals, except as to the limits, are the same as in 
the corresponding formulae for the interior point. 
If in the ^-integrals we put t-\-x 2 — f 2 in place of t, and ultimately suppose x inde- 
finitely large in comparison with/’ they severally become 
f’(<+* 5 -/ i )- i ' +# < ! ’ +s ' , (<+* 2 )“ ! " i to=\’~, SrsSi=* ss " — 
Jo J 0 I'+^j 1 l 2 S + tJ 
yy+^-pr- < !+i (<+^)-*-^=/^p ,=**- ** %!%-* . 
and they all four give 
Ring-integral ; 
. r nsn 
■ x s + 2 q F(i,9 + i)’ 
which agrees with the value 
AYY n(i», *s, is+j, ^5), =~^‘ n(is, is, is+ 2 , oj 
when j is indefinitely large. 
MDCCCLXXV. 
