VOI~ XLIV.] PHILOSOPHICAL TRANSACTIONS. 35Q 



Il.If A, B, C, be any three angles; Z = A + B,X= A— B,H = J-A + B+C. 



Then ^r x v, c-v,x = «, |.c + x x «, fc— x = s, ^a + c — b X s, {-b + c— a =s, iT^ x siT^. 

 And Ir x T),z-t),c =s,^z + cxs, |z"^=s,|a + b +"0 x «, |a + b— c = «,h x s.h^^. 



ni. i^ = — = -IL ::= ^ = ^. or^^3^ =^^ — '"'' — 'Sii _ ^'i^ 

 iV,z rr ~tt',z vfii t\z' «V,|z ~ rr ~ tY ,:^z ~ v ,i ~ 7^' 



v,z t,-^z ~ r ~ sj' 



VI. — = ^'^+ ^'^ _ li£. „„(j /•>- _ ^ _ £if _ *>+«', A _ ^',z X f'.J 

 <,r J, A — «,a t',z ' <,z x <,x t,x t,z *^a— s\A "" rr 



VII. — - = rz r- = - ; if z and x are two arcs, then A = z +x,a = z — ^. 



S,a f,A — «J* SjT _ j; 



VIII. *,r+^ = _i>^X.',^±.-,zXM: ^ ^i±if:.^ 



— r /,2 x/,x 



TV o^ rXT * '^ >< * >^ :+: «, E X «,z rr::f t.z x t,x 



IX. *,^ ± ^ ^: __-__^. 



X. ^z"T^= 4i^'4_rr;andi\r+T=iL5iiiiL^ 



— rr±t,z y.t,x ' — t,z±.t,x i'> 



XI. y,z ± ar = -^;^<^2xf,x '■; and/,z + a; = ^7+7-,- • 



XII. In three equidifFerent arcs a, z, «, where z (= -J-a + a) is the mean arc, ari<l 



X (= -i^A —a) their com. difF. ; put/> = -y, ^ = -^; p = 2.J& X *,z,a = 29 X J,a. 

 Then s,k = p — ^,a = a + «,a ; And ,5, a =: p — 5,a =: s,a — q. 



XIII. Letci=i;,A — v.a = s',a — s',\; then .5*,a — w,a = 2/, a + dy. d = 2s,a—dxd. 



XIV. Let A, B, c, &c. be the sines, a, b, c, &c. the co-sines, a", b\ c, &c. the 

 tangents, of the arcs, a, j3, y, &c. whose number is n; the radius being r; put 

 .« for the product of the n co-sines, /, s", s'", &c. for the sum of the products 

 made of every sine, every two, three, &c. sines, by the other (n — 1, n — 2, n — 3, 

 &c.) co-sines, where the co-sine noted by n — n is unity. 



Then the sine of « + P + y + S,&cc. =/-/"+«'- 5'", &c. X vir.' 



r 



And the co-sine of * + (3 + y + '^,&c. = /-/' + /'-*"' &c. X irr-/ 



r 



XV. Also putting t' for the sum of the tangents of the arcs, «, (3, y, &c. 

 t", t'", t'", &c. for the sum of the products of every two, three, four, &c. tan- 

 gents ; 



Note. — When an arc is terminated in the second, third, or fourth quadrant, some of the signs 

 (+ and — ) of the terms in the preceding theorems, will, by the known rules, become contrary to 

 what they now are. 



