PROFESSOR CAYLEY ON PREPOTENTIALS. 
-T(*.+ ff )r(*- ? ) 
— r(*+ ff )r(4*-gr+i) 
— r(is+g)r.(|-?) 
_ TLsri 
— r(i«-?+i)r(^+y) 
f +1 £ r ? -^+/ 2 -^)-^+/ 2 )~ |,+?-1 ^, 
^ £J_ / /- is - ? (^/ 2 -^) is - ? (^+/ 2 ) ? - f ^ 
where we may in the integrals write t-\-v?—f 2 in place of t, making the limits co , 0 ; 
but the actual form is preferable. 
91. In the third form for f write mf, at the same time changing t into mt ; the new 
value of the disk-integral is 
rioru C” 
Writing her emf=f-\-hf, that is m=l+j, m 3 =l+-y, and observing that if — 
be positive, the factor (m 2 (t-\-f 2 )—x 2 )~ q+ ‘ vanishes for the value t — at the lower 
limit, we see that on this supposition, — 2+i positive, the value is 
=rs r+^K) ? " LS’~ >{t+f - x2+ T ; 
viz. the term in ^ is =$/ into the expression 
that is into 
r (i* + <z)r(i-?) 
which is in fact = of into the value of the ring-integral. 
92. Comparing for the cases of an interior point %<f and an exterior point %>f, the 
four expressions for the disk-integral, it will be noticed that only the third expressions 
correspond precisely to each other ; viz. these are : interior point, k <f; the value is 
r(i*+?)r(j— q) 
(t+f at 
e 1 - 2 ? 
\-q r(is-f-g)’ 
where, if ([ be positive (which is in fact a necessary condition in order to the appli- 
cability of the formula), the term in e vanishes, and may therefore be omitted : and 
