Mr. DE morgan, ON TRIPLE ALGEBRA. 



(A,) m = 0, n =0, q = c = h, 

 a = p, 

 P = a + 2hl. 



But at the same time it is desirable to examine the case of tf = - ^, t" = — ^i the conditions 

 of which are a = -\, 6 = 0, c = 0. These two systems may be called the simple cubic and 

 quadratic systems, both being triple. I now proceed to a mere enumeration of cases to be pre- 

 sently discussed. 



Case A. Let m = 0, n = 0; which gives either of the following 



(J,) m = 0, n = 0, q = - c, 



— Sac - p(c + b) = l(a - p), 



P = a + {b +c)l, 



P= p - 2cl. 



Neither gives a simple quadratic form, unless P = - 1, which is inadmissible. 

 Simple cubic forms are only such as are contained in 



b = c = - q, m = 0, re = 0, 

 al + b{a + 2)) = - 1, 2h(a + p) = - l{a - p), 

 P = a + 2bt = p - 2bl, 

 which give p = a=I, /=— 1, 6 = 0. 



Case B. Let m = 0, n = \. We have then 



ri' = {q + c){q-b)l + br, + cl, 11= ^, 



l- = {(l + c)(q-b)l + cri + bl. ^,, = „. 



This is the case, and the only one, in which the action of ^ upon both of the others is imper- 

 ceptible. The following cases will be considered, the first of which is a species of simple quadratic 

 form, the second a simple cubic, the only one which the case yields. 



Case C. Let m = ^, 



quadratic and cubic forms are as follows : 



ra = 1. This gives 1=0, q = h = c, a = p. 



The only simple 



f = e 



-7?= - 1 +i'? + K' 







Case D. Let m = - ^, w = i. The equations of condition are reducible to 



b + c 



1 = 



The simple quadratic and cubic forms are 



(6 + .3c) (6 - c) = - 2 (a + p). 

 I I2 



