The orthogonal transformation used in a previous paper(1) to derive the distribution of x2 has its counterpart in a double orthogonal transformation which can be applied to a variable y dependent on another variable x in order to deduce the known tests of significance in the simple regression problem. The general theorem is due to R. A. Fisher (2), and the particular transformation now used is a special case of the problem considered by Vajda (3) in the multivariate case. Attention is also directed to papers by Bartlett (4) and Elfving (5). In the present problem the use of two successive transformations is of interest in showing what happens, and in providing a relatively elementary proof of the known results.