Sir Witt1am Rowan Hamirton’s Researches respecting Quaternions. 241 
in which last equation the members are symbols for a numeral set. And thus a 
numeral set (q¢) may come to be conceived as being a system or set of numbers, 
serving to mark or to express the complex ratio which an ordinal set (q) bears 
to a simple or single ordinal relation (a), regarded as a standard of comparison. 
Case of Quaternions ; Coefficients of Multiplication. 
23. In the case of quaternions, the formula (214) gives a system of 4‘ = 256 
equations of condition, included in the following type (in which w has been written 
instead of 7’, and the accent common to all the indices s’ has been omitted as un- 
necessary in the result) : 
1,,14,0 M1, 0,2 Mr, u,1 M18 TH Mp,u,2 Mt,9,2 + Me, 3 Nt, 3,5 
= No, ue Mer,0 7 Mi, u,0 e211 TH Mo, ue Ns 9  N3,u, 5 %t,7,33 (219) 
each of the four indices, 7, s, ¢, uv, in this last formula, being allowed to receive 
any one of the four values, 0, 1, 2,3. And all these two hundred and fifty-six 
equations are satisfied when we establish the following system of numerical values 
of the sixty-four coefficients of multiplication (in which the commas between the 
indices are again omitted for conciseness) : 
No = 13 Nor = 05 Non =03 Mux = 03 
Noo = 0; My = 1; Nn» = 0; Nn3 = 0; (220) 
Now = 03 Non = 03 No. = 13; M3 = 05 
Noo = 05 Nos, = 03 Nose = 05 = M3 = 15 
Moo 0; My = 1; Moo = 0; Nos = 0; 
Myo =—-1l; my =0; my. =O; Ny3 = 0; | (221) 
(Oey = UR 2 te SUS Vea SOR aa Sle 
Mn =O My = 0; “%e— 13 ny, = 0; 
Nisqg =O GPa OF SE gate Lise 172559 = OS 
Ny = O05 Mo, = 05 My2= 03 M,= 1; | (222) 
No =—13 My, = 03 Moyo = 05 No = 05 
Non = 03 Nog, =—13 Mo = 03 Mog, = 05 
Noo = 93 Ny, = 93 Msp = 03 Mo = 15 1 
Nay = 0; Nz, = 03 Nyx. =—13; Ny3 = 0; | (223) 
Ny =O; M3, = 13 N3o = 0; Ny, = 0; | 
Ns =—15 My, = 03 Mage = 03 Ns = 0. J 
