178 



Mr. A. McAulay. 



[Dec. 12, 



If (n) is the reciprocal of (6— n) and (6— n) is the reciprocal of 

 (n), then (u) and (6— n) stand towards 0', the conjugate of 0, in 

 exactly the same way as (n) and (6— ri) stand towards 0. 



The 0' n-tic is the same as the n-tic, and the complex (r) corre- 

 sponding to the root a of the 0' n-tic is reciprocal to (s), (£), . . . . and 

 to (6-n). 



If be self- conjugate (0' = 0) the complexes corresponding to the 

 different roots are both conjugate, with regard to 0, and reciprocal. 



When not one of the roots of the n-tic is zero _1 applied to 

 motors of (n), and supposed to reduce them to motors of («), has a 

 unique intelligible meaning. It is, like 0, a general function. The 

 roots of its n-tic are the reciprocals of the roots of the n-tic. The 

 complex corresponding to any root of the _1 n-tic is the same as the 

 complex corresponding to the corresponding root of the n-tic. 

 1_1 is the conjugate of -1 . 



If is real the coefficients of the n-tic are real, but not necessarily 

 the roots. If a and b are two corresponding imaginary roots (i.e., 

 ab and a + b are real), r = s. The complexes (r) and (s) are then 

 imaginary, but the complex consisting of (V) and (s) is real. In this 

 real complex is included a real complex (2p) of order 2p, where p is 

 any positive integer not greater than r, such that {(0— a)(0— 6)p 

 reduces every motor of (2p) to zero. 



A real general self- conjugate (octonion) -ar differs from a quater- 

 nion self-conjugate, and differs from a Grassmann self-conjugate in 

 that (1) the roots of its sextic (n — 6) may be imaginary, and (2) 

 there may be not more than one motor, A, in the complex corre- 

 sponding to a repeated root a, for which (yr — a)k = 0, whether a 

 be real or imaginary. The roots of an energy function sextic are 

 always real, and when for such a function a is not zero, every motor 

 A of (r) is such that (0 — a) A = 0. But when a = there is not 

 even in this case necessarily more than one motor A for which 

 0A = 0. For a complete energy function a is never zero. 



If degenerates into a commutative function the roots of its sextic 

 are the roots of what was above called the 0i cubic each repeated 

 twice. 



In particular, if degenerates into a self -conjugate commutative 

 function the roots of the sextic are all real, and consist of three 

 pairs of equal roots. The three corresponding complexes are three 

 sets of coaxial motors, whose axes are three mutually perpendicular 

 intersecting lines. Except when what were called the principal 

 roots of the cubic are all ordinary scalars, there are not in this case 

 six co-reciprocal motors, which also form a conjugate set. 



Combinatorial variation is applied to prove some other facts, w 

 being a general self -conjugate, and (n), as before, the sum of the 

 reciprocals of the pitches of SF 1 wF 1 ,SF 2 wF 2 , .... where F^Fo .... 



