170 



Mr. A. McAnlay. 



[Dec. 12, 



increase mentioned ; and T 2 Q is an " additor," i.e., it effects the 

 addition-increase mentioned. The amount by which the pitch of the 

 motor operand A is increased is called the pitch of the octonion 

 operator Q. When a motor B is itself thus considered as an operator 

 its versor is a quadrantal versor and its translator is unity, i.e., it 

 does not translate the operand at all ; hence the twister of a motor 

 is a qnadrantal versor. The tensor of B is the magnitude of its rotor 

 part, and the amount by which it increases the pitch of the operand 

 is its own pitch. 



These results are all established by aid of quaternions on a purely 

 Euclidean basis. They have, of course, mechanical interpretations 

 in connection with (1) the instantaneous motion of a rigid body, (2) 

 a system of forces, (3) the momentum of a system of moving matter, 

 and a system of impulses. The corresponding motors are called 

 velocity motors (Sir Robert Ball's K twist on a screw "), force motors 

 ("wrench on a screw"), momentum motors, and impulse motel's 

 ("impulsive wrench "). If A and B are two force motors A-f B is 

 the force motor of the system of fornes obtained by the composition 

 of the two systems corresponding to A and B ; and similarly for the 

 motors of the other types. 



The octonion operator Q( )Q _1 is exactly analogous to the qua- 

 tericn operator q( )q~\ It displaces the octonion operand in the 

 most general manner as a rigid body; it translates the operand 

 parallel to the axis of Q through a distance double of that through 

 which Q when regarded as a motor operator translates the motor ; 

 and it rotates it as a rigid body round the axis of Q throngh double 

 the angle of Q. If A is the velocity motor of a rigid body which in 

 the time t has suffered the displacement Q( )Q _1 , 



A = 2}IQQ-\ 

 and if R is any octonion fixed relative to the body, 



R = MAMtt. 



The geometrical connections between A. B (two motors), MAB. :nl 

 SAB are examined. The axis of MAB is the shortest distance 

 between A and B. Its rotor part bears the same relation to the rotor 

 parts of A and B that the vector Xxfi does to the vectors * and /3. The 

 pitch of MAB is the sum of d cot 6 and the pitches of A and B, where 

 d is the distance and 9 the angle between A and B, d being reckoned 

 positive or negative, according as the shortest twist which will brin? 

 the rotor of either A or B into coincidence, both, as to axis and 

 sense, with the rotor of the other is a right-handed or left-handed 

 one. 



A scalar octonion such as SAB requires two ordinary scalars, S a AB 

 and .sAB to specify it. S AB bears to the rotor parts of A and B the 



