Here is a (sketch!) answer to 1.47.
We have e(p,u) = k(u)g(p) which is in the "Hicks" variables (p,u) whereas the income elasticity is defined on the "Marshall" variables (p,y). I shall use the Theorems connecting Hicks and Marshall together with the Chain Rule.
From Theorem 1.7.7 p.43,
de(p,u)/dp = xh(p,u)
so xh = kdg/dp
and dxh(p,u)/du = k'dg/dp.
From Theorem 1.9.1 p.43,
x(p,y) = xh(p,v(p,y))
and so x = kdg/dp (*).
Also dx/dy = dxh/du.dv/dy.
From Theorem 1.8.1 p. 40, e(p,v(p,y)) = y so de/du.dv/dy = 1 and since
de/du = k'g,
dv/dy = 1/(k'g).
Thus dx/dy = (1/g)dg/dp (*).
Also y = kg (*).
Income elasticity equals y/x.dx/dy and the result follows on substituting the expressions marked (*).