238 Sir Witr1am Rowan Hamiton’s Researches respecting Quaternions. 
successive derivations are performed ; it is evident that we can then proceed, in 
like manner, to perform on the resulting set a third successive derivation; and 
that, with respect to such successive operations of derivation, the following simple 
but important formula holds good : 
Kip Rees Reker (205) 
To develope this symbolical equation, which may be said to contain the asso- 
ciative principle of the multiplication of numeral sets, we may conveniently 
employ a characteristic of numeral separation, N, analogous to those two charac- 
teristics, M and rR, which we have already introduced in this paper, for the pur- 
pose of expressing separately the different moments of a momental set, and of 
separating, in like manner, those constituent ordinal relations between moments 
which compose an ordinal set. Let us, therefore, agree to regard the m equa- 
tions, 
iM, = Neg 5 HS Nyy ss Me = (206) 
as jointly equivalent to the one complex equation or expression (154), for a 
numeral set g, of any proposed order 7; in such a manner that we shall have, 
identically, for numeral constituents and numeral sets, the equations 
My Na (Mm, mM. 2 Nn, =); } (207) 
Mm, =N,(™, My -. Mn), --- 
and 
q = (Nolo Nie +> Naa) (208) 
which are analogous to those marked (10) and (11), for moments and momental 
sets, and also to the formule (57), (58), for constituent ordinal relations, and for 
the ordinal sets to which they belong. We may then substitute for the formula 
(153) of symbolic multiplication, or of successive derivation, the following : 
N;- Ke Kr = 1, 7,83 (209) 
which will give, also, by suitably changing the letters, 
Nye Xe Xe = M5,¢3 (210) 
the commas in the indices being here, for the sake of greater clearness, restored. 
In this manner we find that 
Nei Xi Kes +) = z, . Ny vt, s Ne, e, 5! (211) 
