308 NOTE ON THE FUNCTION *tX = or (c — x) -r- {c (c — x) — If}. [687 
Suppose that a, b, c are the sides of a triangle the angles whereof are A, B, C; 
then c 2 = dr + & 2 — 2ab cos G, or we have 
(A+I> ! = 4 00S «c ; 
A, 
or, writing this under the form 
V(A) + ^7^ = 2 cos G, 
the value of A is at once seen to be = e 2iC ; and it is interesting to obtain the 
expression of the nth function in terms of the sides and angles of the triangle. 
The numerator and the denominator are 
V l P + Q, 
\ n R + S, 
where 
P = A {acc + /3) + (— 8x + /3), R — A {yx + 8) + yx - a , 
Q = — {ax + /3) — A (— Sx + /3), S — — (yx -f 8) — A (yx — a). 
Hence, writing the numerator and the denominator in the forms 
\i n P + A-* n Q, 
A* w R + A"* 71 S, 
these are 
(P + Q) cos nC + (P — Q) i sin nC, 
(R + S) cos nG + (R — S) i sin nG; 
viz. they are 
(A - 1) (a + S) x cos nG + (A + 1) {(a -8)x+ 2/?} i sin nG, 
(A — 1) (a + 8) . cos nG + (A + 1) {2y# — (a — 8) } i sin nC, 
or, observing that ^ = i tan G and removing the common factor i (A +1), they may 
be written 
tan G (a + 8) x cos nG + {(a — 8) x + 2/3} sin nG, 
tan G {cl + 8) . cos nG + {^yx — (a — 8) } sin nG. 
Substituting for a, /3, y, 8 their values, these are 
tan G {(c 2 — a 2 — 6 2 ) x cos nG) + {(b 2 — a? — c 2 ) x + 2a 2 c} sin nG, 
tan G {(c 2 — a 2 — b 2 ) . cos nG) + {— 2cx — (6 2 — a 2 — c 2 )} sin nG, 
= tan C {— ab cos Gx cos nG } + {— ac cos B .x + a 2 c } sin nG, 
tan G {— ab cos Gx cos nG } + {— cx + ac cos B } sin nG, 
= x {— ab sin G cos nG — ac cos B sin nG) + a 2 c sin nG, 
— cx sin nG + {ac cos B sin nG — ab sin G cos nG) ;