72 
ME. W. H. L. EIJSSELL ON THE CALCULUS OE SYMBOLS. 
§7r + V — 2§ V+ §(7r^ + tt) + — |( 7r + 1 ) + 
— g‘^(7r^+27r) + §(7r^+7r) 
— f^(7r^-l“27r) — §(77 
^ 77 ( 77 + 1 )^-!"’^* 
§77(77 + 1)^ + ’*'^ 
The results of the four last examples may be written thus : 
{q-\-7r)~^(q^-{-2q^(7r + l) + §77(277 +l)+77®)=§^+f7r+’r^ 
(§77 + 77^)-(^V+2) + ^^(277+3)-§(377^-h377 + l)-h77^}==§^-*^(77-fl)4-7r‘^ 
(§® + §^(27r + l) + §(27r^+2x + 1)4-77'' )(g4-x)“* = g^+^77+7r^ 
(§ V — 2 § V + § (77“ + x) + 77 ‘‘)(g 77 + 77 ^) ~ — 1(77 + 1 ) + 77 ^ 
I now come to tAVO propositions of great importance. 
First, to determine the condition that §\|/j(x)++(x) shall divide the symbolical func- 
tion 
+ f "'‘P„-i(7r) + f <p„_2(x) + &C. + g(pi(77) + <po(x) 
internally ^vithout a remainder, 
where the symbolical quotient is 
^ +( 77 — 1 )'^ I \I/i(7r — 1) \I/i(7 r— l) 4 -i(x— "[+( 77 — 1 ) 
4'i(77 — ' \I/,(77 — — 2)4'i(77 — 3 )^"^ J 
The required condition is found by equating the remainder to zero ; and we have 
I l/o 77 vt/o( 77 -l)+( 77 - 2 )... + ( 77 -n 4 -l) . — O 
— — 2)4'i( 77 — 3) ...l/j(7r — w)™”! ^ ’ 
where ^%^,(x) + -4/o(’r) is internal factor of f<Pn{'^)+f‘ '?’«-i(’r)+ &c. +(Po(t). 
Hence we see how we may resolve the symbolical function 
+ + + +f'Pl(^)4'‘Po(^ ) 
into factors in all possible cases. 
