1895.] 



Octonions 



171 



same relation as the scalar bears to the vectors x and fi. sAB is 

 most simply described mechanically. If A is the velocity motor of 

 a rigid body on which the force motor B is acting, — sAB is the rate 

 at which the corresponding system of forces is doing work on the 

 body. — *AB is, therefore, the product of the tensors of A and B, 

 multiplied by what Sir Robert Ball calls the virtual coefficient of the 

 two corresponding screws. If A and B are rotors, sAB is + six times 

 the volume of the tetrahedron which has A and B for a pair of oppo- 

 site edges. sAB/SiAB is what is called the pitch of the scalar 

 octonion SAB. It is the sum of — d tan and the pitches of A 

 and B. 



If, A, B, C are three motors, S ; ABC bears to their rotor parts 

 the same relation that the scalar Sxfi-{ bears to the three vectors 

 x, /J, 7. The pitch of SABC is the sum of d cot 0— e tan 0, and the 

 pitches of A, B, and C ; where d and are related to A and B, as 

 before, and where e is the distance, and the angle between C and 

 the shortest distance of A and B. 



The rotor part "MjABC of the motor MABC bears, as to direc- 

 tum and magnitude the same relation to the rotor parts of A, B, C 

 as the vector Vxf3-( bears to the vectors x, ft, 7. The pitch of MABC 

 is the sum of (e tan (p—d cot 0), (cot 2 6 tan 2 9 + cot 2 + tan 2 0) and 

 the pitches of A, B, and C. 



Similarly as to the rotor Ml(MAB)C of the motor M(MAB)C. The 

 pitch of M(MAB)C is the sum of d cot + e cot and the pitches 

 of A, B, C. 



A finite motor, whose pitch is infinite, is called a lator. [The 

 term " vector " is not here used, because though lators and vectors 

 have the same fundamental geometrical properties, and obey the same 

 laws of addition, they do not obey the same laws of multiplication, 

 for the product of two lators is always zero.] Thus every motor 

 consists of a rotor part and a parallel lator. If Q is an octonion, 

 MiQ stands for the rotor of the motor of Q and mQ for a coaxial 

 rotor of the same maguitude and sense as the lator of the motor of Q. 



If MAB = 0, either A and B are coaxial, or one of them is a lator 

 parallel to the axis of the other, or they are both lators. If M t AB = 0, 

 either A and B are parallel, or one of them at least is a lator. If 

 oi AB = 0, either (p + p') sin 6 -f d cos = (where d and 6 are as 

 before, and p and p are the pitches of A and B), or one is a lator 

 parallel to the axis of the other, or they are both lators. 



If SAB = 0, either they intersect perpendicularly, or one is a lator 

 perpendicular to the axis of the other, or they are both lators. If 

 SAB = 0, either they are perpendicular, or one of them at least is a 

 lator. If *AB = either (p + p') cos 9 = d sin 6>, or one is a lator 

 perpendicular to the axis of the other, or they are both lators. 



The necessary and sufficient condition to ensure that SABC = is 



N 2 



