312 Proceedings of Royal Society of Edinburgh. [sess. 
But, while <£<77 and merely repeat the type £7, and £7^ 
and the type we have novel forms in the combinations 
c7 <£, an( ^ ‘^i4 > v 
Many of these are useful in the solution of equations among 
forms ; such as, for instance, 
X 2 = ^ x l / X == X'l / '> or x I/X = X'l / > &c * 
where x is to be found when i Jr is given. One simple result of the 
above discussion, which is often of great use in such matters, is the 
obvious condition that two such forms shall be commutative in 
their successive application. 
4. When two roots are imaginary, all the forms above are still 
real ; since, when (3 and y take the forms (3 ± iy, y and z must be 
written y ± iz. In the forms (&), the imaginary terms cancel one 
another; in (a) the real terms do so, and the whole is divisible 
by i. 
5. Of course, with a, f3, y (as in 2. above) and three scalar 
constants, we can produce any form of linear and vector function. 
And the paper concludes with forms in which these constants are 
merged in a new arbitrary vector. 
