1895.] 



Oct onions. 



175 



degree. Two motors, E and F, are said to be conjugate with re- 

 ference to utT. when sEc<rF =,0. In the language of " Screws " they 

 are conjugate motors of the complex of the second degree. -nrE is 

 what Ball calls the polar of E. 



When sr = 1 conjugacy reduces to reciprocity, and every theorem 

 relating to the former becomes one concerning reciprocal motors. 



A real motor, E, is called a positive, negative, or zero norm, according 

 as sE-orE is —1, -f-1, or zero. When nr = 1 a positive norm has positive 

 pitch, a negative norm negative pitch, and a zero norm zero or infinite 

 pitch, i.e, it is a rotor or lator. (n) can always be expressed in an 

 infinite number of ways as a complex of n real independent conjugate 

 norms. The number of norms of any type (positive, negative, or 

 zero) is definite, and the complex of zero norms is a definite 

 complex. 



If (n) be a given complex of order n, and (m) a given complex of 

 order m included in (n), another complex (w— m) of order n — m can 

 always be found with the following properties, (m) and (n—m) 

 together make up (n), so that (n — m) is also included in (n), and 

 (m) and (n — m) are independent, (m) consists of the positive norms 

 Ai A 2 . . . . , the negative norms Bi B 2 . . . . , and the zero norms 



d Co.,.. Di D-2 (n—m) consists of the positive norms 



A x ' A 2 '. the negative norms B/ B 2 ',. . . . , and the zero norms 



C/ C 2 '. ... D/ D 2 ' The number of the norms Di D 2 . ... is the 



same as the number of the norms D/ D 2 ' All pairs of these 



norms are conjugate except the following pairs of zero norms (DjD/), 

 (Do Do') These last are such that 



sDxarD/ = sDtsrDz = . . . = — 1. 



The pair of motors Ax", B/', deduced from any such pair of excep- 

 tional zero norms D 1; D/ by the equations 



A/' = (Dx-f D/)/^, B/' = (Dx-D/)/y2, 



are conjugate, and are respectively a positive and negative norm of 

 (n). Thus (n) consists of the following set of conjugate norms : 



(1) positive, Ax , A/ , Ax". . . . , (2) negative, B x . . . . , B/ , 



B/'. . . . , (3) zero, C x . . . . , Cx' (n) and (n—m) are called semi- 



conjugate complexes. 



Denoting the complex of A x A 2 . ... by (A), that of A x A 2 . ... 



Bx B 2 by (AB), &c, it is proved that (A'B'CC'D) contains all 



the motors of (n) which are conjugate to (m) or (ABCD), and no 

 others. (A'B'CC'D) is therefore called the conjugate of (m). The 

 sum of the orders of (m) and its conjugate exceeds n by the order 

 of (C), which is the complex common to (m) and the (definite) 

 zero norm complex (CC) of (n). The conjugate of the conjugate 

 of (m) is not in general (m) itself. 



