Vol. XliVIII.J PHILOSOPHICAL TRANSACTIONS. 473 



— 2 X (a + w)" X (a — «)' + (a —«)*■= 4 X (2 + 2q) X (2 + 1r), &c. X 



■-' X ("'+ rrj «') X ('-* + 7^^a'), &c.jbbeing=l(art.2)and|^a'(=i') = 0. 

 Moreover, one of the other cosines q, r, s, &c. being = — 1 (art. 2) some one 

 of the factors 2 + 29, 2 + 2r, 2 + Is, &c. will vanish ; which factor being ex- 

 punged from the product 4 X (2 + 2q) + (^ + 2r), &c. and restored to the 

 divisor w* + — ^^ a^, or u^ + —^ a^, &c. from which it was taken, that divisor 



will become Aa^; and the product 4 X (2 + 2^) X (2 + 2r), &c. will then (by 

 art. 5) be = 'rp. 



Consequently (a + w)^" — 2 X (a + w)" + (a — w)" + (a — ^y, will be = 

 n^ X u^ X 4a"^ X (w^ + c'O X (m^ + rf^), &c. where the factor Aa^ takes place in- 

 stead of (0^ 4* sq. of the tang, of 90°. 



If 3/ be = 1, and n an odd number, p will be = 1, and i = O; but no one 

 of the cosines q, r, s, &c. will be = — 1 , as when n is an even number. There- 

 fore, in this case, the equation (a -f- w)** — 2y X (a + u)" X (a — u)" -|- 

 (a - u)"" = (2 -f 2y) X (=-' + b') X («' + c"), &c. becomes (a -f- a.)*" - 2 X 

 (a -I- 0,)" X (o - «)" + (a - a,)'" = 4 X 0," X (a,' + c") X (o.^ + c/'O, &c. 



^rt. Q. By taking the square root of (a + uY" — 2 X (a + «)" X (a — «)- 

 -(-(« — w)'", and of its two values just now found, we have, when n is an even 

 number, (a -\- u)' — (a — w)" =: 2ania X ^ 1/ -\- c- X Vm^ + <^^ &c. 2a taking 

 place instead of v^ u'^ -\- sq. of the tang, of 90". 



And, when n is an odd number, {a -{- u) — (a — u) = 2u X ^w'^ -|- c' X 

 v' u'^ 4- d.^^ &c. Whence the following construction is inferred. 



^rt. 10. Describe, about the centre c (pi. 11, fig. 1 and 2), with the radius 

 a, the circle pa' a' a'", &c.; draw the diameter pca, and the tangent b"'pb*; 

 divide the semicircumference pa'g into as many equal parts pa', a'a", a"a"', &c. 

 as there are units in the integer n; draw the secants ca'b', ca"b", &c. and, 

 taking on ca any point o, draw k"'ok* parallel to b'"pb*; likewise draw b'k', 

 b"k", b"'k"', &c. parallel to pq, and call co, a. 



Then will q be the cosine of twice the angle pca', r the cosine of twice pca", 

 i the cosine of twice pca"', &c. if the radius be 1 . 



Therefore pb' := ok' will be = c, pb" ^ ok" = d, &c. and ok' = V^^ Hh~?, 

 ck" = ^ u^ -^ d-, &c. Consequently op" — oq" bein g = (a -j- w)" — (a — u)", 

 and n X pa X CO X ck' X ck", &c. = 2anu X /IF+~? X /w^-f d\ &c. 

 when n is an even number; op" — oa" will then be = n X Pa X co X ck' X 

 ck", &c. where the diameter pa takes place instead of the infinite quantity ck*'. 



But if n be an odd number, op" — oa" will be = 2 X co X ck' X ck" x 

 ck", &c. 



^ri. 11. It is evident that, of the factors ck', ck", ck"', &c. the first and 

 last, the second and last but one, &c. are resj)ectively equal to each other. 



VOL. X. 3P 



