ME. W. SPOTTISWOODE ON THE CALCHLHS OE SYMBOLS. 
101 
For the fourth degree, 
)t - — bxx + 3^;") 
3%’^" + 3%' V + % " 
O o ' 2 O 
'^—X 
— ^ ' 
^xy+{^xx — 3%'>+ to' — %") 
(:^' — 3% )’r" + (%" — 3 %%')t+ ( to — 3% ') 
and 
— (%' — ^XX + 3%")t- to + ^x^+xx—x' 
— {x^—^xx+ 3%" )t — + 5%'% - 3%% ' 
+ 3%%' ' — % " 
x' - 6% V - 4'/'%" +^x^—x'=—x{—x^+ ^xx —%')+(—%'+ 3%>:' — x)' ; 
« 
or if Kj, Ea, . . be the remainders of ‘r(T+%)' 
‘r^(‘r+%)“*, . ., we have 
And generally, if 
then 
^2=— 
1^3= — Xi4^2 + R'2» 
E4= — ;;(^E3+E3. 
(’*■ + %)“’= Qn+ + 
the remainder of which must be contained in the last term. Performing the actual 
division, and remembering that 'rK„=E„T+E'„, 
+E'„(R„ 
R„t+R„% 
— %P«+K- 
Hence we have generaUy, 
I^n+i= 
and consequently, remembering that Ro=l, we have the condition that 4i(f)'^+'4^o(g’) 
may be an internal factor of • • ?’o(f)? 
where 
'Po(?)Po + ?’l(g’)Ri+ • • <P»(f)Iln— 0, 
R.. 
4^o(g) 
'l'l(g) 
( 4 .) 
The law of the quotients is best seen by actual division. In case of 92^v^+?>i7r+(Po, 
given above, the quotient may be written 
