paraphrase:
1) Equation 1 is essentially an equation for point conditions. On the other hand, the experiment with the melting ice is a transient one. Consequently, it can be used to demonstrate the validity of equation 1 only if the proper integration procedures are used. The rate of heat transfer at any instant is given by equation 2.
If it is assumed that the ice melts at a constant temperature of 0°C and that the measured temperature of the surroundings does not vary by more than about 1°C in the course of the experiment, then it is justifiable to assume that ∆T remains constant. However, both h and A will vary as the sphere melts since both are dependent on D. Equation (3) may be obtained by constructing an energy balance for the ice sphere: