472 PHILOSOPHICAL TRANSACTIONS. {[anNO 1754. 



Consequently n' is = 4 x (2 + Iq) X (2 + 2/) X (2 + 2*), &c. when the 

 factor, whose value is nothing, is expunged. 



^rt. 6. — Let us now suppose y = — 1, and n an odd number: then one of 

 the cosines p, q, r, &c. being = — 1 {Art. 3), 



^2n _ m^_^ + w'li' will be = {v" + 0) X [v" — 1 + 1p X v + 2p) 



X (t^' — 2 + 1q X t^ + 2 + 29) &c. Therefore, dividing by v^, 



'■ — 2to''"-' + -\- n^ will be = (t;* — 2 + 2j& y. v + 1 -\- ip) X {v^— 



2 + 2q X V + 2 + 2q), &c. and consequently n^ = (2 + 2p) X (2 + 2q) X 

 (2 + 2r), &c. when the factor, whose value is nothing, is expunged. 



Art. 7. — Substituting in the equations z'-" — 2j/z" 4-1=0, and 2^ — 2xz + 



1=0, (— — ) instead of 2, we have 



^O — u' 





and 



(^-^—Y—lx X — 1- i = (" + ")'- ^■^ X (" + ") + ("-'") + (" - ■" )' 



^ a — «' a — « (a — »)* 



1 — J' 



= ■ r i-^ — = O. Consequently, 



(a — «)* ^ -^ 



(a + 0,)'" - 23/ X (a + o))" X (« - ")" + (a-«r will be = (2 + 2p) X (2 + 2^) 



X (2 + 2;), &c. X (^.^ + —^ a') X («' + ~f a^) X (o,^ + —^ a^), &c. 



But, by Art. 4, (2 + 2p) X (2 + 2q) X (2 + 2r), &c. is = 2 + 2j/, the 

 upper or lower of the two signs prefixed to y taking place, according as n is an 

 even or an odd number. 



Therefore (a -f «)'" — 2y X (a + «)" X (a — u)" + {a — w)" is = (2 + 2i/) 



X ("' + ^-f-«^) X (<-♦ +~^- «^)X {.' + — ->^), &c. 



Now/) being the cosine of any number of degrees, radius being 1, a^ will 



be the square of the tangent of half so many degrees, radius being a : therefore, 

 denoting that tangent by b ; and the tangents of half the arcs described with the 

 radius a, whose cosines, when the radius is 1, are q, r, s, &c. being denoted by 

 c, d, e, &c. respectively ; we have (a + u)" — 2j/ X (a + w)" + (a — u)" + 

 (n - 6,)'^" = (2 + 2y) X (<o^ + b') X (=0^ + c') X (o.^ + cP), &c. But when y 

 is =: I, and n an even number ; or_y = — 1, and n an odd number; 2 + 2y 

 being = O ; nothing can be determined from that equation : therefore, in those 

 cases, recourse must be had to what is done above. 



^rt. 8. — Let us suppose j/ = i , and n an even number : then the equation 

 (^a + <.)'"-27/X{a+^yx{a-coy+{a-^y"={2+2p)X{2+2q)X{2+2r),kc. 



X (c,^ + i^-^ a') X i'^' + ~f a^) X (c-^ + \~ a^), &c. becomes (a + .y 



