PROFESSOR CAYLEY ON PREPOTENTIALS. 
ATI 
viz. the right-hand side is here equal to the left-hand side or is =0, according as 
. . . — |— < 1 or >1. V is consequently obtained by multiplying the right-hand side by 
j ' l 
dx . . . dz and integrating from — oo to -fco for each variable. 
Hence, changing the order of the integration, 
V : 
' orVIL 
irT^s + q) 
r 
du dv dr v ls+q r ls+q l Qu . O, 
where 
Now 
if 
Q=J&...*cos|^m— e\— yt + T (( a — *)*■•■+(«— *) , })»+2(i«+2)*|- 
f i+<a -«y=i*£z r + ^ ... r +T ^, 
&—X— f* Ta 
% 1+/V 
j, n h\c 
... 
158. Substituting, and integrating with respect to £ ... £ between the limits — oo , -f oo 
we have 
(/. ..h)t 0 S (/ 2 ^ C 2 T \ I 1 ) 
Q= (i +7^.. : i +^)V- C0S ir~^ T ~n7V --r+v;) b+ M : 
or, what is the same thing, writing i in place of r, this is 
n (f...h)-jfi s t* U a 2 c 2 e 2 \ . i ) 
Q= t/4<...tf + qw- c ” {\ u -rTr-wTrj) v+ ^\’ 
that is, writing 
have 
a 1 c _i_e 
a ~f r +t"^w+i^ r T 
v =T®j , . , n>** 
or, writing 7r s_1 =I (F^) s , this is 
TT 
— f dt ■ f dudv . fl ? cos{(%— <r)v-\-\qr\<pu 
A VS S + 99 Jo Vjo Jo 
159. Boole writes 
du dv d q cos\(u— (?)v-\-\q7r\<$>u= <p(<r) ; 
viz, starting from Foukiek’s theorem, 
^ du dv cos(w — <r)v . <pw=<p(<r) 
(where <p(c ) is regarded as vanishing except when <j is between the limits 0, 1, and the 
limits of u are taken to be 1, 0 accordingly), then, according to an admissible theory of 
m; 
t~ q ~ l v q cos{{u— <r)v + ^q7r}<pu . 
