By PANTAZOPOULOS K

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F. of 0 is simply given by F(0) = ^ Î0+ 2 £ {ap sin/>0 + j8p(l - cos/>0)}//>! 8) is valid by Jordan's test (Titchmarsh, 1958, p. 406) provided /(0) = H / ( 0 + o) + / ( 0 - O ) } . 2 THE CHARACTERISTIC FUNCTION 43 Further, by Dirichlet's test (Titchmarsh, 1958, p. 407), the representation is also valid if /(0) has only a finite number of maxima and minima and a finite number of discontinuities in the interval (0,2π). 3. Independence and Convolution Let θχ and 02 be two angular random variables.

If the data is grouped, we shall assume that there are k class-intervals of lengths h. Then 0f will denote the mid-point of the /th class-interval, f{ the observed frequency in the /th interval and n the sum of the frequencies. In our terminology, we regard descriptive measures as statistics. This approach helps not only to study their population counter-parts but also to remind us that we are dealing with sampling variations. FIG. 1. Representations of the /th sample point 9t. OP = 1. 2. 1. ,xn' represent the same observations when the distances are measured from 0' instead of 0 along the x-axis, put 00' = a.

We now indicate some applications. Let x0' and S0' be the values of the mean direction and circular variance after a shift of a. 15) so that, given the values of x0' and S 0 ', 3c0 and 5 0 can be obtained. 4 can be used when the components are required relative to a fixed direction a. 3), 1 — C will measure the devia tion in this direction. 4. CALCULATION OF THE MEAN DIRECTION AND THE CIRCULAR VARIANCE With the help of four typical examples on grouped data, we illustrate how x 0 and S 0 can be obtained for different situations.