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an amplitude; and thus we are led to write, generally, as a 
transformed expression for a versor, 
Ug = cos Z Q + WQ.sin Z @. (25) 
The amplitude of a vector is in this theory a quadrant; 
that of a positive number being, as usual, zero, and that of a 
negative number two right angles. Applying the same prin- 
ciples and notation to the case of the continued product 
aBcpa of the four successive sides of an uneven quadrilateral 
ABCD, we find that the amplitude Z ascpa of this quaternion 
product is equal to the angle of the lunule ancpa, if we em- 
ploy this term “lunule” to denote a portion of a spherical 
surface bounded by two ares (which may be greater than 
halves) of small circles, namely, here, the portion of the sur- 
face of the sphere circumscribed about the quadrilateral ancp, 
which portion is bounded by the two ares that go from the 
corner A of that quadrilateral to the opposite corner c, and 
which pass respectively through the two other corners B and 
p. The tensor and scalar of the continued product of the 
four sides of the quadrilateral do not change when the sides 
are taken in the order, second, third, fourth, first; and gene- 
rally, 
cos Z Q = SQ + 1TQ; (26) 
so that we have the equation, 
Z ABCDA = Z BCDAB; (27) 
hence the two dunules ancpa and BcpaB, which have for 
their diagonals ac and Bp the two diagonals of the quadri- 
lateral, and with which the lunules cpaBe and DaBeD re- 
spectively coincide, are mutually equiangular at a and B. 
Thus, generally, for any four points, ABCD, the two circles 
ABC, ADC cross each other at a and c (in space, or on one 
plane), under the same angles as the two other circles, BCD, 
BAD, at B and D. 
a = FyPdigs a: Be ie - 
