662 
Proceedings of the Royal Society 
Area of Complete Plane and Total Volume of Elliptic Space. 
The area of a biangle having the infinitely small angle a is 
>7r p 
r 
pal S] 
*S0 
pa/ sin - dr = 2 p‘ 2 a . 
~ 7 0 p 
Hence P = 4 tt P 2 =— ... (6) 
7T 
From this result we can deduce very easily the total volume S of elliptical 
space (single). The locus of the most distant points on the radii through any 
point of space is a plane. Suppose this plane divided up into infinitely small 
regular quadrilaterals (squares) of side k. The volume dS contained by four 
radii drawn to the vertices of one of these figures is 
Hence 
( 7 ) 
This curious result can also be obtained by calculating the volume swept out 
by a complete plane rotating through 180° about any line in it. 
Formulae for Right-Angled Triangles. 
Let ACB (fig. 18) be a triangle right angled at C. Let BA& = <£A. , B b=da. 
C&A=B + t£B. If Bm be perpendicular to A b, then bm = dc. 
We have at once by (3) 
sin B da = p sin - dA. . 
• • (8) 
P 
Also 
dc=da cos B 
• • (9) 
Calculating the area BA6 in the two different ways we get 
Whence 
i. e. 
9 
l 
D dc — p 2 e . 
dkp 1 - cos I'j = p\dk + dB) 
dB = - cos — d . A 
( 10 ) 
