81 



sin (2«7r + ^), i being any integer number ; so also in this cal- 

 culus, 



cotr [^, x] = cotr [</> + 2 ir, x - ^ «V] 

 tres [^, x] = tres [^ + 2iV, x~2^V] 



where r = -^. The quantity r holding in the present theory 

 the same place that tt does in the calculus of sines. From the 

 theorem of Moivre we learn that 



(cos^ + V^^^. sin(^)"' =: cos »z(^ + 2i7r) 4- V^ — 1 • sin m (^ + 2i7r). 

 The corresponding theorem already announced by Mr. 

 Graves may be written as follows : 



(cotr[</., x] + ttres[^, x] + i'tres[x,^])™ 



= cotr [m(^ + 2iV), »i(x— 2jV)] + itres {m{^ + 2iV), ?w(x— 2^V)] 



+ litres \m{x — ^ir), m (^ + 2^■^)]. 



Among the most important consequences flowing from the 



preceding theorem is a mode of resolving an equation of the 



form 



z^—pz^ + g'Z"--^ = 



into its cubic factors. For this purpose we must first reduce 

 the equation to the form 



y^ — Zcoiv{A, b)?/'-^" + 3cotr(— A, — B)r/»— 1 = 0. {b) 



Now it will be found that if we multiply together the three 

 expressions 



a? — cotr [^, X] - tres[^,x]- t''es[x,^] 

 X — cotr [0, x] — a tres [^,x] — "^t^^s [x? ^] 

 X — cotr [^5 x] — a^tres[0,x] — a tres [x, ^] 



in which a stands for ~ , the cube root of -f- 1, the 



product will be ) ^- >< 



a;''— 3 cotr [^, x] x^ + 3 cotr [ — ^, — x] ^— 1 • 

 Hence it may be proved that each of the cubic factors of {b) 

 will be of the form 



