174 



Mr, A. McAulay. 



[Dec. 12, 



type. Making this fact the basis of a classification, it follows that 

 there are (rejecting the case = 0), thirty-four different types of 0. 



If >jr is a commutative self -conjugate, such that SiE^-E is negative, 

 and not zero for all motor values (except lators) of E, there are always 

 three (and generally only three) real lines which are not all parallel 

 to one plane such that any three real motors with these lines for axes 

 form a fully conjugate set both with regard to and 0. 



A commutative self-conjugate is never a complete energy function, 

 and can only be a partial energy function when it degenerates into a 

 lator function. 



The properties of the general function are examined by the help of 

 " Grassmann's Ausdehungslehre." Motors are in Grassmann's own 

 geometrical interpretations quantities of the second order. " Inner- 

 Multiplication, " and the thory of " Normals," have in this case inter- 

 pretations which depend on an arbitrarily chosen origin. Hence 

 Grassmann does not apply his theories concerning these two to 

 motors ; but motors may be treated as quantities of the first order, 

 and if they are, and if Sir Robert Ball's meaning of " reciprocal " is 

 identified with Grassmann's meaning of " normal " a real motor with 

 negative pitch is a "simple imaginary" quantity of the first order, 

 and the " numerical value " of a lator or a rotor is zero. Grassmann 

 generally assumes his quantities to be real and always assumes that a 

 quantity is zero when its numerical value is zero. Hence nearly all 

 his theorems require modification in our case but his methods are 

 generally with some extensions applicable. 



" Combinatorial Multiplication " receives several applications. A 

 particular scalar combinatornal product of six motors is much used 

 and also a motor combinatornal product of five motors. The latter 

 is the motor (with a certain definite tensor) reciprocal to the five 

 motors of the product. By aid of these two products, the following 

 results (the first of which is contained in Ball's " Screws ") are estab- 

 lished. To every complex of order n there is a complex of order 6 — n 

 reciprocal and no motor not belonging to the latter, is reciprocal to 

 the former. If (w) be a complex of order n, and (6 — n) an indepen- 

 dent complex of order 6— n, then if (6—^) is the complex reciprocal to 

 (n) and (n) the complex reciprocal to (6 — n), (n) and (6 — n) are 

 independent complexes. 



What is called " combinatorial variation " is a species of variation 

 of which Grassmann's "linear" and "circular" variations are 

 special cases. Particular cases of combinatorial variation are these 

 two, and also "hyperbolic" and "conjugate" variations, the last 

 being with reference to a given general self- conjugate function. 



Let nr be a given real general self-conjugate function. The motors 

 E of a given complex (n) of order n, for which sE^rE = from what 

 Sir Robert Ball calls a complex of the (n— l),the order and second 



