THE INDEX SYMBOL IN THE CALCULUS OF OPERATIONS. 
20 
where 
.• (i-p+ 1 ) ^ ih'-I) . . (i-p) 
1 . 2 ..P ’ ^^p+>~ 1 . 2 .. 
(P+1) ’ 
so that A and B are two consecutive terms in the system of the degree i. Then opera- 
d d 
ting upon with A, and upon with B, in the. same way as was done in the case 
of the third degree, we have 
4 d d'^'^ 
As.- • • Si -r-. — 5 — - 
*dy P * ^dx'-PdyP+^ 
ji 
d d^^^ * 
B^i j-=Nb . iSS, ..Si^, — 
^ dx P+1 ' 'dx^-Pt 
But 
and 
^dx^-PdijP+^ 
H-Np+iS 5 , — 1 )< 
d' 
— I.3..(p-hl) ’ 
in other words, Np- 1 -Np+, is equal to the (pd-l)th coefficient in the case of the degree 
(2 + 1 ). Hence, adding the above written expressions for As,. B^i^, and calling the 
value of Np+Np+, ,+,Xp+,, we have 
=+(*4 -’ft)+®( 4 -+' 
But if it is true (and it has been proved in the cases of 2 , 3 , . . ) that 
A=SM(w,~a,) .. (2q_p — (^— ^;)a._p)(2;i_^+, — (2-^ + 1 )/ 3 ,_p+,) .. -(2— 1 )/ 3 ,_,), 
B = S22(2q-a,) .. (2q_p_,-(2-^-l)a,._p_,) (2;,-_p — (i-i>)A_p) • • ('Wi-i — 5 
i. e. if A and B are respectively equal to the sums of all the products of the above forms 
that can be formed by interchanging the 22s and vs, so that the total number of the 22s 
and of the vs remains constant in each product (viz. (2 — ^jp + l) 22-factors and {p — 1 ) 
v-factors in A. and (2 — p) 22-factors and p v-factors in B) ; then will A(Vi — 2 / 3 ;) + B(22, — 2a,) 
MDCCCLX. F 
