81 
sin (2ir + ), ¢ being any integer number; so also in this cal- 
culus, 
cotr [o, x] = cotr [ + 2ir, xXx- 2ir] 
tres [¢, x] = tres [p+2ir, x—2ir] 
where 7 = The quantity r holding in the present theory 
the same place that 7 does in the calculus of sines, From the 
theorem of Moivre we learn that 
(cosp+V eee de sing)” =cosm(p+2ir)+V —1.sinm (p+ 2i7). 
The corresponding theorem already announced by Mr. 
Graves may be written as follows : 
(cotr[#, x] + etres[¢, x] + ?tres[y, p])™ 
=cotr[m(+ 2ir), m(y—2ir) |] +ctres [m(p + 2ir), m(y—2ir)] 
+ Ptres[m(x — 2ir), m(p+ 2i7)]. 
Among the most important consequences flowing from the 
preceding theorem is a mode of resolving an equation of the 
form 
2" —p2z™ 4. g2"—r = 0 
into its cubic factors. For this purpose we must first reduce 
the equation to the form 
y"—3cotr(a, B)y™" + 3cotr(—a, —B)y"—1=0. (0) 
Now it will be found that if we multiply together the three 
expressions 
«— cotr[p,x]— tres[¢,x]— tres[y, ¢] 
x — cotr[, x] — atres [p, x] —a’tres[y, 6] 
a —cotr[ 9, — a°tres[, x] — atres [y, ¢] 
4/3 
2 
in which a stands for — , the cube root of +1, the 
product will be } 2- x 
2° —3cotr[p,x]2? + 3cotr[ —¢, —y]a—1. 
Hence it may be proved that each of-the cubic factors of (6) 
will be of the form 
