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an amplitude; and thus we are led to write, generally, as a 

 transformed expression for a versor, 



UQ — cos Z Q + WQ . sin Z Q . (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 

 ABCDA of the four successive sides of an uneven quadrilateral 

 ABCD, we find that the amplitude Z abcda of this quaternion 

 product is equal to the angle of the lunule abcda, if we em- 

 ploy this term " lunule" to denote a portion of a spherical 

 surface bounded by two arcs (which may be greater than 

 halves) of small circles, namely, here, the portion of the sur- 

 face of the sphere circumscribed about the quadrilateral abcd, 

 which portion is bounded by the two arcs 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 

 D. 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 -^ TQ ; (26) 



so that we have the equation, , 



-^ ABCDA = Z BCDAB ; (27) 



hence the two lunules abcda and bcdab, which have for 

 their diagonals ac and bd the two diagonals of the quadri- 

 lateral, and with which the lunules cdabc and dabcd 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. 



