Sir Wittram Rowan Hamiton’s Researches respecting Quaternions. 218 
indicated by those characteristics. Suppose then that any ordinal quaternion q, 
or any set of four ordinal relations, a, b, c, d, between moments of time, is pro- 
posed as the subject of the operations. 
For the purpose of operating on this quaternion by the characteristic of deri- 
vation 7, we may first write the following definitional equation between its two 
symbols, 
qi (anb;.c;d)): (73) 
and then resolve this complex equation into its four components, or constituents, 
with the help of the signs of ordinal separation, r,, &c., as follows : 
HEC ask gd — bis TRG) e5" RAqi— 1. (74) 
In the next place, the definition (70) of 7, combined with the notation (66), 
directs us to change the signs of the second and fourth of these equations (74), 
and then to make the first and second equations change places with each other, 
interchanging also, at the same time, the places of the third and fourth, so as to 
form this new system of four equations : 
RG —— Dis) Regia eRe qi = —1d 55 Rg) =e! (75) 
We are then to combine these four constituent ordinal relations, thus partially 
inverted and transposed, namely, — b, a, —d, and ¢, into a new ordinal quater- 
nion; and this will be, by definition, the first coordinate derivative, iq, of the 
proposed quaternion q; so that we may now write, as derived from the equation 
(73), by the first coordinate mode of quaternion derivation, the equation, 
iq = (—bja, — d,c). (76) 
If now we repeat this process of derivation, we get successively the two following 
systems of four equations : 
Rtg — bs B,.2q)== a; B,.1qg = — ds. B,.8q Ses (77) 
Rey. eq SS — "8 3) Rgetg! SS bs BL QV =e; agg —d5i\, (78) 
and, finally, by a new combination of these four last ordinal relations into one 
ordinal quaternion, which is the derivative of the derivative of q in the first co- 
ordinate mode, we find 
?q = idiq = (—a, —b, —e, —d)= —q; (79) 
