176 Mr. A. McAulay. [Dec. 12, 



When {n) and (m) and nothing else is given the statements of the 

 following table are established. 



The following com- 

 plexes are deter- 

 minate. 



The following complexes are arbitrary to the extent 

 mentioned. 



(CC) 



(CCD) 



(CD) 



(A'B'CCD) 



(ABA'B'CCD) .... 



(C) is any complex which with (C) makes up (CC) 



(D) „ „ (C) „ (CD) 

 (AB) „ „ (CD) „ (ABCD) 



(A'B') „ „ (CCD) „ (A'B'CCD) 



Also (D') is indeterminate to the extent implied by the state- 

 ment: — (DD') is any complex which contains (D), is conjugate to 

 (ABA'B'), and with (ABA'B'CC) makes np O). The individual 

 motors Oi, C 2 . . . . may be taken as any independent motors forming 

 the determinate complex (C), and the individual motors D 1? D 2 . . . . 

 may be taken as any independent motors which with Ci, C 2 . . . . form 

 the determinate complex (CD) ; similarly for d', C 2 \ . . . ; but not 

 similarly for D/, D 2 ' 



The table is much simplified, (1) when ■& is an energy function and 

 (2) when nr — 1 and n — 6. 



If tjr is any self -conjugate and E is one of a set of six conjugate 

 motors, such that sE-srE = 0, then tsrE = 0. If nr is an energy function 

 (partial) and E is any motor, such that sE-srE = 0, then w~E = 0. 



w (a general self- conjugate) can always be expressed in an infinite 

 number of ways in the form -srE = — 2/FsEF, where F is one of six 

 real constant independent motors, and / a corresponding ordinary 

 scalar, which may be zero. The six motors which are reciprocal to 

 each set of five of the F's form a conjugate set. When none of the 

 /'s are zero -sr -1 has a unique intelligible meaning. The F's form a 

 conjugate set with regard to w" 1 . 



When (n) and (m) are given the numbers of motors of the different 

 types A, A', &c, are all definite numbers characteristic of •sr. There 

 are thus when n = 6, two methods of classifying -sr. The number of 

 types of -ar (rejecting the case w = 0), when classified according as 

 it affects motors in (m) only, is 2, 5, 9, 14, 20, or 27, when m has the 

 value 1, 2, 3, 4, 5, or 6 respectively. The number of types of 

 -ar, when classified according as it affects motors in (in) and in a semi- 

 conjugate complex (6— m), is 27, 77, 125, 145, 125, 77, or 27, when m 

 has the value 0, 1, 2, 3, 4, 5, or 6 respectively. 



(n) being as before, v being a given general self- conjugate, and Y r 



