172 



Mr. A. McAulay. 



[Dec. 12, 



either two independent motors of the complex A, B, C are lators or 

 XA + YB + ZC = 0, where X, Y, Z are scalar octonions whose 

 ordinary scalars (S, X, &c.) are not all zero. 



If B and C have definite not parallel axes, XB + YC is any motor 

 that intersects the shortest distance of B and C perpendicularly, 

 where X and Y are arbitrary scalar octonions. 



The analogue in octonions of the linear vector f unction of a vector 

 in quaternions is called a commutative linear motor function of a 

 motor. It is not the most general form of a linear motor function of 

 a motor. The latter is called a general function. If a commutative 

 function is such that, acting on an arbitrary rotor through a definite 

 point, it reduces the operand to a rotor through the same point it is 

 called a pencil function, and the point is called the centre of the pencil 

 function. The geometrical relations between a pencil function and 

 rotors through its centre are precisely the same as the geometrical 

 relations between a linear vector function of a vector and sectors. 

 When a commutative function degenerates into a lator function (for 

 all values of the motor operand) the geometrical relations between 

 the function and rotors in general are also precisely the same as the 

 corresponding quaternion relations. A general function involves 

 thirty-six ordinary scalars, a general self -conjugate twenty-one, a 

 commutative function eighteen, a commutative self-conjugate twelve, 

 a pencil function twelve, and a self -conjugate pencil function nine. 

 [In the case of the pencil function three of the scalars go to specify 

 the centre.] «r, a general self-conjugate, is called an energy function 

 when sEtsrE is not positive for any motor E ; it is a partial or complete 

 energy function, according as sEw-E is zero for some values of E or 

 for none. 



We here pass over for the most part those properties of the com- 

 mutative function which are immediately deducible from the fact that 

 octonions are formal quaternions. 



Let for the present stand for a commutative function. satis- 

 fies a cubic with scalar octonion coefficients. This cubic generally 

 has three definite scalar octonion roots ; but sometimes it has an in- 

 finite number, sometimes only one, and sometimes none at all. The 

 cubic (called the 0! cubic), whose coefficients are the ordinary scalar 

 parts of the coefficients of the cubic, has an important bearing on 

 the geometrical properties of 0. 



can always be put in a trinomial form analogous to the corre- 

 sponding quaternion trinomial form. 



If X is a root of the cubic corresponding to a single root of the 

 0! cubic, 0— X can be put in a binomial form, and there is a line 

 such that if E be any motor coaxial with the line 0E = XE, so that 

 0E is coaxial with E. If X corresponds to a repeated root of the 0, 

 cubic, this is not always true. 



