168 Transactions of the Boyal Society of South Africa. 
Put 7r - «j = ^x, and we get 
Snca^z , , o . , o^^ /^^sin^ 2 
/•^ sin^ 2x cos 2x ^ 
o 
Let us now consider the two integrals 
0 
= ^|(2 -~ A;2)K - 2E| from equation 20. 
/2sin^ 2x . cos 2x 
T 
sin^ 2y — 8 sin^ y cos^ x 
= Ii - 81,, 
sin^ Y cos 2 Y 7 
where I, = / ■ ch- 
0 
For the evaluation of I.^ we note that 
dx^ Ax ^ ~ ^'X ' 
Integrating, we get : 
0 o 
Again, for the reduction of 
d ^sin'^ X cos _ k'^ sin'^ x cos^ ^ 3 sin"^ x _ 4 sin^ x 
J "i'X .J Av ^ J Ax 
sin-^ X 
^X 
0 
we have by integrating the identity 
r • X o7 9 sin^ Y , ^..sin^x i 1 
-|- (sm X cos X . ax) = 3/.^ . — ^ - 2 (1 + ^^-^^ + ^ 
J Ax ^ AX 3/.^ • 
7 (K - E), 
o 
p sm^ X cos^ X ^S-k' p- sm^ x ^ 
J a3y ^ ^x ■ ^Sk^ ' 
But 
Ax 
