Significance tests for the introduction of one new function or two orthogonal new functions to represent a series of measures have previously been derived as approximate formulae. The exact formulae have now been reduced to single integrals involving only the number of observations and one function of the measured values, and their evaluation has been carried out. The approximate formulae are as accurate as we need when the number of observations exceeds the number of unknowns by about 6, but the extended calculation also gives the results when the difference is smaller. An approximate treatment is given for the case where the normal equations for two harmonics are not orthogonal in the coefficients and the weights unequal. This is the usual practical case. The solution for the symmetrical case, which has been done accurately, can be adapted to the general one by including two extra factors, of which one is calculated and the other can easily be found in any given problem.