PROEESSOR CAYLEY ON PREPOTENTIALS. 
741 
or, instead of (J“©)', writing J'©+J m ©', this is 
□ V=-^^(J’_2P+J)_j^-(0'0 t -00' o ). 
We have here 
( i 2 1) fl 21) 
J'-2P+J=« 2 j (fl + , + ^*-(fl + , + /s)(fl + ys) + (T[7^)-- • +e2 j(9 + #“(0 + rj)fl + fi 5 } 
=tf- 
=jj 2 . Q, suppose. 
Also ©'©„—©©„' contains the factor jj, is = ??M suppose. 
106. Substituting for J, J 7 — 2P+J, and ©'© 0 — ©@ 0 ' their values 77P, ??Q, and j?M, the 
whole result contains the factor n m+ \ viz. we have 
f t 
Kfl+ZW + i+Z*)* 
(fl + AW + ’J + ^ 2 )‘ 
□ V=- 
4»j w + 1 P” 
and if here, except in the term t] m+ \ we write q=0, we have 
“ 4- e2 -J 
2’ 
u 
W+fT' 
~(fl + A 2 ) 2 
a 2 
(fl + /2)4*' 
r ‘ 2 
. . + £ 
^(0+A 2 ) 4 
e _ 1 T w 
‘ M ’ — 6^0 > 
»-i-« r ® 
M=0 O 0 O "-0 O ' 2 , 
and the formula becomes 
□ v= -4^ +1 J o ,ni - 2 |iJ o '"0 o + J 0 '(©o"-^) j ; 
or (instead of J 0 , 0 O ) using now J, 0 in their original significations, 
J=l- 
this is 
or, what is the same thing, 
□ V = - 4y m+1 J te - 2 |l J'"0+ J' j, 
-A 2 ) 2 ^fl 2 J 
(9 + A 2 )' 
viz. the expression in { } is 
— f a? ■ C 2 , c 2 ] . x r a « c 2 , e 2 -| f 1 
L(0 +/2j4- (fl + A2)4t- fl4 J f 2 + y 2)2 - * ’ "T ( fl + A 8jS -T g 2 J |_( fl + ^2) 2 - 
We thus see that q being infinitesimal □ V is infinitesimal of the order t] m+1 ; and hence 
jj being =0, we have 
□ V=0; 
5 p 2 
