ME. W. H. L. EUSSELL ON THE CALCULUS OF SYMBOLS. 
71 
The following are instances of internal multiplication : — 
^ — TT 
— P'TT — 
f + § — w" 
§T — TT^ 
— ^T(7r + 2) 
— §77^ +T^(t+ 1) 
g>^(T + l)V^ — g'T(^ + ^r+ 2 ) + T^(‘T+l)- 
The results of the four last examples may be written thus 
(^T — — (t + 1)) = (g>V — g>(T^ + TT + 1) + 7r)T(T + 1 ) 
— (■r+l))(g’T — ‘;r^)=(g>^(5r^+7r) — T+2)+T)T(‘r+l). 
I shall now give some examples of external division, the divisor being a polynomial. 
I> + ^(tt + 1 ) d" 2 TT -j- 1 ) + + giT + T* 
g^(Td-2) 
^*^+^■^( 2 ^+ 1 ) 
g'Vd-^T( TT + l) 
f7r4-’r^)f^(^+2)4-f*(2Td-3) — ^(35r^d-37rd-l)d-’r‘‘(f^— ^(^4-1) + ^* 
g-(T+2)+g^( r+2r 
— f{ ’^+1}* — f(3'r^4'3'^+l) 
-f{ T+i)^-g( TT+iy 
§71^ -{-T* 
We shall next consider some examples of internal division. 
f + 7r) d- §"(27r d- 1 ) d" t (27r" d- 2 TT d- 1 ) d- (§" + d- TT* 
^^-\-o^7r 
^ ^ ( TT -j- 1 ) d" t ( 2 TT* d" 2 TT d- 1 ) 
g"(^d-l)d-g7r'-^ 
§(7rd-l)"d-’r" 
g(7rd-lfd-^" 
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