178 Transactions of the Royal Society of South Africa. 
If I = sin^ja^ + i f?a^ + i, 
a — o 
then we get from the reduction formula 
-2)...3 ^(p 
IS 
odd. 
(p - 1) (p - 3) . _ - r/l'v 12 ■ " P " 
Or we may write 
I,, = Ios+i- J, _ (2s + l)! 
= I./ 
Hence, if n is odd (n = 2m + 1). 
27r'» . r2'" + i «-i 22-''+i (s!)2 '"-1 (2s) ! 
V„ = + 1 - 2»7. + r' {2s + 1)T 2^-^7«T)2 
= 2^>iin 28Tr ' 
Again, if n is even (n = 2m) 
[2(m-l)! , --2 1 
= b 77'" 7'2w i-i V /ox 
2'"-i . m [(m - 1) !]2 ' * 2s + 1 
Again we note that the area of a circle in space of -dimensions is equal 
to the volume of a sphere in (n — l)-dimensions, and the circumference 
of the circle in /i-dimensions is equal to the surface of a sphere in (n — 1)- 
dimensions. 
Of, therefore, 
V„ = volume of sphere in 7?.- dimensions, 
S„ = surface of sphere in ?i-dimensions, 
Aw = area of circle in ?i-dimensions. 
Cw = circumference of circle in ?i-dimensions, 
