474 VHILOSOPHICAL TRANSACTIONS, [aNNO 1754, 



Therefore, omitting the squares of the factors below pa, and the squares of their 

 values, OP" — oa" is = w X pa X co X ck'* X ck"^ X ck'"^ &c. and {a + «)" 

 — (a — «)" = 2anu X (w' + c^) X (w^ + dr), &c. when n is an even number; 

 or OP" — oa" is = 2 X CO X ck" X ck"' X ck"", &c, and (a + oi)" — (a — w)" 

 = 2w X (w'^ + c^) X (u^ + d,'^), &c. when n is an odd number. 



^rt. 12. If we suppose y = — I, and n an odd number, it will appear, by 

 proceeding much in the same manner as in art. 8, that {a -\- u)*" + 2 x (a + w)" 

 X{a- «)- + (a - u,T is = n^ X 4a^ X (co" + /-') X (o'-* + c^) X i<^' + d') &c. 

 where the factor 4a' takes place instead of w' + sq. of the tang, of 90°. 



If 3/ be = — 1, and n an even number, (a + uif" + 2 X (a + w)" X (a — «)" 

 + (a - w)*» is = 4 X («" + i') X («' + c-"), &c. 



Whence, by extracting the square root of both sides of those equations, we 

 have, when n is an odd number, (a + u) -\- {a — a>)" = Ian X ^ u^ -\- b" X 

 V u^ -\- c% &c. la taking place instead of ■v/j^^ -f. sq. of the tang, of QO"; and, 

 when n is an even number, {a + w)" + (a — w)" = 2 X ^^I^b"- X V^w^ -f ^, 

 &c. Hence we infer this construction. 



Art. 13, Having described about the centre c (lig, 3 and 4) with the radius a, 

 the circle va'K'a"x", &c. draw the diameter pca, and the tangent b"¥h*; divide 

 the semicircumference pa'a into as many equal parts va', a k, x'a", &c. as there 

 are units in In; draw the secants ca'b', ca"b", &c. and, through any point (o) in 

 ca, draw k"oh* parallel to b"sb*; likewise draw b'k', b"k", &c. parallel to pa; and 

 call CO, u. 



Then, if the radius be 1, /> will be the cosine of twice the angle pca', q the 

 cosine of twice Pca", &c. ther efore vb ' = ok' will be = b, vb" = ok" = c, &c. 

 and ck' = -^^J+l-^ ck''= ^ u,'' +7=, &c. 



Consequently op" + pa" being =_(«_+ ")" + (« — ")"j and ?2 X Pa X ck' X 

 ck", &c. = 2an X V''^'^ -\- b"^ X /w' + c*, &c. when n is an odd number; op"+ 

 oa" will then be = ra X pa X c/4' x ck", &c. where the diameter pa takes place 

 instead of the infinite quantity cA^""*"*- 



But if n be an even number, op" + oa" will be = 2 X c^' X ck'\ &c. 



Art. 14, It is obvious that, of the factors ck', ck", &c. the first and last, the 

 second and last but one, &c. are respectively equal to each other: therefore the 

 squares of the factors below pa, and the squares of their values, being omitted, 

 OP" + oa" is = n X pa X ck'^ X ck"\ &c. and (a + «)" + (a — «)" = 2an X 

 (a« _|- /,«) X (u" + c^), &c. when n is an odd number; or op" -|- oa" is = 2 X 

 ck' X ck'\ &c. and (a + «)'" + (a - i.)" = 2 X (co' + Z-') X (<.' + c'), &c. when 

 n is an even number. 



Art. 15. Writing, in the equation (a + wf — 1y X {a ->{- w)" X (a — «)" + 

 (o — !o)'" = (2 + 2y) X («* + i') X («* + c»), &c. (found by art. 7) a — m for 

 u, the same becomes (2a — «)*" — 2j^m" X (2a — u)" + w'" =. (2 T 2y) x 



