76 
MACFARLANE. 
what different from that of Hamilton, as we shall thus en¬ 
tirely avoid the use of Cartesian coordinates. For this pur¬ 
pose we write S . a p = — d a , where a is any unit-vector, the 
meaning of the right-hand operator (neglecting its sign) 
being the rate of change of the function to which it is ap¬ 
plied per unit of length in the direction of «. If a be not a 
unit-vector, we may treat it as a vector-velocity, and then 
the right-hand operator means the rate of change per unit of 
time due to the change of position. Let a ft y be any rect¬ 
angular system of unit-vectors, then 
V — + yd y 
which is identical with Hamilton’s form.” 
In the investigation which follows I use Greek letters and 
also ij k to denote axes of the real unit sphere, and the same 
preceded by \/— 1 to denote axes on the imaginary unit 
sphere. Minus p denotes a unit-vector pure and simple, 
while j/— I p denotes Hamilton’s quadrantal versor. The 
rules of reduction for the real unit-vectors are: 
i* = l;f=l; ]c 2 == 1; 
ij — V— lk;jk= i/—1 i; ki= i/—1 j; 
j'i — — V — 1 k ; kj — — j/— 1 i ; ik — — j/— 1 j. 
It follows that the rules for the imaginary unit-vectors are: 
(!/=! i) i) — 1 j 
(i/—ij) (i/ 13 !i) = — i; 
Cl/=M k) (j/-—1 id) — — l. 
(i/—i i) (V— li)^— V—(V— ij) (V 7 —10 = ]/—l & ; 
(l/— Ij) (v— lk)=— j/— 1C* (l/— 1 F)(l/— Ij) = l/—Ti; 
(y—lk)(i/—li')~ — y^j. (}/ — 1 i) ( j/ 1 k) = f/—T j. 
The order is assumed to be the order of writing. 
