STUDIES IN STATISTICAL REPRESENTATION. 485 
Now, to convert the length 1 into —1 a geometrical con- 
vention has been adopted that this may be done by two 
operations: the first being effected, it is in a position at 
right angles to the original position: the second operation 
again places it at right angles to the new position, and it 
is then in the negative direction as regards its first position: 
the first position is regarded as the geometrical represen- 
tation of the operation denoted by i or ;/ —-1. ? 
This suggests the suitability of a similar convention for 
log -—1. Since log1=0 we have to deal with a point 
instead of a straight line of unit length. Suppose a closed 
curve symmetrical with respect to the ¢ axis, (a circle will 
do for simplicity) in a plane perpendicular to the xy plane, 
and with the origin as its lowest or highest point, according 
as the curve is above or below the plane. A point, moving 
round this closed curve from the origin, returns again to 
the origin after one complete circuit. The operator : signi- 
fies that it is in a new position, viz., that which it would 
have after passing through the angle z. This is consistent 
with the fact that +. = 1. As the origin represents log 1 
we must have log —1 represented by the point attained in 
half a complete circuit: that is, it will be on the other 
extremity of the diameter, somewhere on the ¢ axis. 
Consequently, if log 1 be represented by the origin, log 
—1 may be represented by two points on the < axis equi- 
distant from the xy plane, one above and one below. The 
convention may be extended so as to give the points a 
definite position. For, writing down the identity —1 =. 
cos 7 + i sin 7 or —1 = e*7' we see at once that the 
* It is erroneous to say the operator 1 rotates a line through 37, because 
t would be +7 and 21 would bez. Also, it may be argued that wis -1, 
not 1”, and thereforei is not really ~-1 except by a mere convention. 
The essence of the matter is that i is an operator, not a multiplier, and in 
the calculus of operations it is not established that f¢ is $2, where ¢ is 
any operation, though ¢” may be used to denote d¢...repeated ton times. 
