309 
687] 
NOTE ON THE FUNCTION ÒX = Ct 2 (c — x) + {c(c — x) — b 2 }. 
or, writing herein b sin G = c sin B, these are 
— acx {sin B cos nC + cos B sin nC] + a 2 c sin nC, 
— cx sin nC + ac {cos B sin nO — sin B cos nC}, 
whence finally 
A' l x ■ 
a 2 sin nC — ax sin (nC + B) 
a sin (nC — B) — x sin nC 
As a verification, writing n= 1, we have 
a? sin G — ax sin A 
Ar = 
a sin (G — B) — x sin G 
sin A 
ax — acx 
sin G 
sin (G-B) 
ac ^—- — cx 
or observing that 
ac 
sin G 
sin ((7- B) 
sin G 
= c 2 — b‘\ 
(for this is sin A sin (G — B) = sin 2 G — sin 2 B), we have 
a 2 (c — x) 
c 2 — b 2 — cx 
as it should be. If in the formula for % 1l x we write x = 0, we have a formula given 
in the Senate-House Problems, January 14, 1878: it was thus that I was led to 
investigate the general expression.