1818.] Mr. Adams's Solutions of Equations. 349 



Article VI. 



Solutions of Equations. By James Adams, Esq. 



(To Dr. Thomson.) 



SIR, Stonekouse, Dec. 23, 1817. 



Your inserting the following solutions, &c. in your Annals of 

 Philosophy, will much oblige Your humble servant, 



James Adams. 



A solution of the equation 2 cos. m A = z ra + — on a sup- 

 position that 2 cos. A = z + \ . By putting the cos. A = c, 

 and - = x we have 



z 

 2 c ~ Z + -T 



4 c 2 = (z + x) 9 = z 9 + x 9 + 2 

 8 c 3 = (2 + x) 3 = z 3 + x 3 + 3 (z + x) 

 16 c 4 = (z + x) 4 = z 4 + x* + 4 (z 9 + a s ) + 6 

 32 c b = (z + x) 5 = z 5 + x 5 + 5 (z 3 + x 3 ) + 10 (s + x) 

 64 c« = (z + x) 6 = z 6 + x 6 + 6 (z 4 + x 4 ) + 15 (z* + x 9 ) + 20 



&c. 

 From whence we get 



2 +x= 2c =2 cos. A 



z s + x 4 = 4 c 9 — 2 =2 cos. 2 A 



2 3 + x 3 = 8 c 3 - 6 c =2 cos. 3 A 



z 4 + x 4 = 16 c 4 - 16 c 9 + 2 =2 cos. 4 A 



z 5 + x 5 = 32 c 5 - 40 c 3 + 10 c =2 cos. 5 A 



z 6 + x 6 = 64 c 6 - 96 c 4 + 36 c 9 - 2 = 2 cos. 6 A 



a- + x m = z» + ^ = 2» . c- - m 2— ■ c— * + -^HT- 



0m _ 4 ™_ 4 «(m-4)(m -t) ._ s ..„ . «(m-5)(m-6)(m -7) 

 * • c "~ P. 2 .3 1.2.3.4 



2 m ~ B . c"- 8 - &c. 



A well-known series for twice the cosine of a multiple arc 



when radius is unity. Therefore 2 cos. m A = z m + — . 



A solution of the equation (cos. A ± V — 1 sin. A) m = cos. 

 m A ± *f — 1 sin. w A may now be readily effected. 



From the preceding equations we have 

 z« - 2 cos. A . z = - 1, and z* m - 2 cos. to A . z m = - 1. 



By completing the squares, &c. 



2 = cos. A + V cos. 4 A — 1 = cos. A + V — 1 sin. A 



