642] 
33 
[641 
the arc of meridian 
-cosp . 
— < p, viz. this 
sm|) 1 
642. 
ON A DIFFERENTIAL RELATION BETWEEN THE SIDES OF A 
QUADRANGLE. 
X 
more than this, we 
r which k is greater 
3, viz. this may be 
it may be greater 
outh of the equator, 
leyond the foregoing 
; sin -1 ^, which may 
r value than 360°. 
iition, which in its 
lircle or half-ellipse 
inding half-diameter 
dius of the original 
[From the Messenger of Mathematics, vol. vi. (1877), pp. 99—101.] 
Let the sides and diagonals YZ, ZX, XY, OX, OY, OZ of a quadrangle be 
f g, h, a, h, c, and let the component triangles be denoted as follows : 
A = AYZ0, =(b,c,f), 
B — AZXO, =(c,a,g), 
C=AXYO, = (a, b, h), 
£l = AXYZ, = (/, g, h), 
Z 
viz. A, B, 0, il are the triangles whose sides are (b, c, f), (c, a, g), (a, b, li), (/, g, h) 
respectively, so that il = A + B + 0. Then we have between (a, b, c, f, g, h) an equation 
giving rise to a differential relation, which may be written 
il (Aada + Bbdb + Cede) — (BOfdf+ OAgdg + ABhdh) = 0. 
This may be proved geometrically and analytically. First, for the geometrical proof, 
it is enough to prove that, when a and b alone vary, the relation between the in 
crements is Aada + Bbdb = 0; for then a and g alone varying, the relation between 
5 
c. x.