104 
ME. W. SPOTTISWOODE ON THE CALCTJLHS OE STI^IBOLS. 
which also may be transformed as follows : — 
( 2 .) 
A similar process of division will be found, in the case of ^^= 3 , to lead to the follow- 
ing remainder : — 
•Po ~ ‘P) + P 2 PT + (“Pl 2^2 + Spsj) + (p2 ^Pl)^ 
4 'oV 
'J'l 
4^1 
(«)(«)+(«)■}. 
( 3 .) 
If ■'Fa, . . represent the ^/-functions, coefficients of the (ps in this expression, the 
law of their formation will be found to be as follows : — 
U/- 
X 1 i ? 
Vi 
•F3 
And generally we may write 
And if El, Ej, .. represent the remainders in the cases ofw=l, n= 2 , .. respectively, 
we have 
Ei=(Po-p;+p,’Fi, 
E'2=Po Pi+P2+(Pl ^p\y^ l-\-p^2-> 
1^3 = Po Pi + p2 P 7 + (Pl 2<p2 + 3 p 3 )'Fi + ((p2 393)^2 + Pa’'!^ 3 ; 
whence 
E2=Ei+?5"-2p'2-Fi+p2-^^2, 
E3 = E2-<p'3"+3p'3'-Fi-3p;-^2+p3^3. 
With a view to forming the expression for E„, let the symbol affixed to E,- signify 
that in the expression for E„ the suffixes of the (p^ have all been increased by unity. 
Then, by the principle of division. 
