140 ON THE MECHANICAL DESCRIPTION OF CERTAIN SEX TIC CURVES, 
that is 
a 2 + b 2 — c 2 + d 2 + 2ab cos (a — /3) + 2ad cos (a — 8) + 2bd cos (/3 — 8) = 0 ; 
or, substituting herein the values 
1 — u 2 . 2 u n 1 — v 2 . n 2v 
cos a = ■=——-, sin a = , cos /3 = = , sin /3 = = „, 
1 + u 2 1 + u 2 1+v 2 1+v 2 
this is easily found to be 0 = 
1 
2u 
u 2 
(a + 
b) 2 — 2 (a+b)d cos 8 + d 2 — c 2 
2 ad sin 8 
{a - b) 2 + 2 (a — b) d cos 8 + d 2 — c 2 
2bd sin 8 
2 ab 
2bd sin 8 
(a — 
b) 2 — 2{a — b)d cos 8 + d 2 — c 2 
2ad sin 8 
(a + b) 2 + 2 (a + b) d cos 8 + d 2 - c 2 
or writing, for greater convenience, 8 = 0, that is, taking cl, /3, 7 to be the inclinations 
of the sides a, b, c to the fixed side d, this is 0 = 
(a + b -d) 2 — c 2 
0 
(a — b + d) 2 — c 2 
0 
2 ab 
0 
(<%-b-df-c 2 
0 
(a + b + df — c 2 
{A + Bu 2 ) + 8abuv + v 2 (C + Du 2 ) = 0, 
where, writing for shortness 
a + d — b — c = A, — a + b + c + d = \', 
b+d — c—a = fi, —b+c+d + a = /jf, 
c + d — a — b = v, —c+d + a + b=v', 
a + b+ c+ d = o-, —d + a + b+ c— p' m } 
A = — vp , B = A//, G = A'/i, D = v'cr.