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S0 MUCH statistical work is performed with approximate data that this writer has long held the opinion that approximate methods also are frequently justified, both for treating the data and for rough checking purposes. Again, where approximate solutions will serve, these may, in many instances, be obtained readily and easily by graphic methods, particularly where the functions considered will plot to straight lines, making interpolation easy.2 Hyperbolic, or reciprocal, grids are a relatively unknown and, for that reason, rarely used type of graph upon which certain functions will plot either as a continuous linear curve or as a broken series of linear curves. For rate studies and for purposes of obtaining weighted arithmetic averages graphically, the hyperbolic grid will frequently render great service. The justification of the present paper is that nearly all standard texts on graphics seem to ignore the hyperbolic grid entirely, or give it inadequate treatment. Perhaps the best generally available explanation of this graph and its usage occurs respectively in the graph paper catalogs of Keuffel & Esser Company, Inc., and of the Codex Book Company, Inc.; and, in the latter, references are made to two articles and to a Codex leaflet which further describe the uses of the hyperbolic grid. To whom should go the credit for having devised this type of graph seems to be unknown, for the present author has nowhere been able to note reference to its originator. Ready-prepared hyperbolic graph papers, conventionally on paper 8X l in size, may be obtained from manufacturers of graph papers, and, on these commercial papers, rulings are sufficiently numerous to provide for reasonably accurate plotting and interpolation. The horizontal axis, or axis of abscissas, is calibrated according to the reciprocals of the numbers printed on this scale. The vertical scale, or axis of ordinates, is customarily graduated arithmetically, and the numerical values given to these graduations may be large or small as desired. The vertical scale is graduated from zero (0) in both positive and negative directions, if desired; the horizontal scale conventionally begins with unity (1) and is calibrated positively to the right to infinity (co).

Lord recently published a nomograph “for calculating a multiple correlation coefficient (R 1.23) from the zero-order correlations (r 12, r 13, and r 23).” His paper reminded us that in 1947 we developed a graphic method of computing R 1.23. If the sole object is speed of computation, the advantage would doubtless lie with Lord's nomograph rather than with our graphic method. But we might claim two possible advantages to the method outlined in this paper. First, the computation can be carried out on ordinary graph paper. Second, we believe that our graphic method can help many students to understand the basic relationships between the zero-order correlations and the multiple-correlation coefficient.

IN PAST YEARS many methods have been developed for fitting polynomials. Although the use of orthogonal polynomials for fitting continuous functions probably originated with Legendre, Tchebycheff2 seems to be the first to use orthogonal polynomials for fitting discrete observations with and without equal intervals. Orthogonal polynomials have been the subject of many mathematical dissertations and an entire volume3 of references and cross references on this subject has been compiled by a committee, of which J. Shohat is chairman. The fitting of polynomials by the use of orthogonal functions has been mainly by two methods: (1) summation; and, (2) multiplication of the variates by actual values of the orthogonal polynomials as given in a table. Pearson4 and Isserlis,i dealt with the fitting of non-equidistant and unequally weighted data. R. A. Fisher developed methods of calculation of Tchebycheff polynomials by successive summation up to the 5th degree,6 and in his classic work on "Yield of Wheat at Rothamsted"7 refers to work by Esscher.8 F. E. Allan published a paper, "The General Form of Orthogonal Polynomials for Simple Series."9 C. Jordan'0 develops the theory in full, and crediting the method to Tchetverikoff in a paper published in Russian in 1926, demonstrates the building up of general expressions for the orthogonal polynomials in factorial form originally used by G. F. Hardy in "Graduation of British Offices Tables, 1863-93." Jordan also suggests the numerical method of building up the polynomials by summation and furnishes tables for this purpose. J. Shohat gave the mathematical approach in "Stieltje's Integrals in Mathematical Statistics."" A. C. Aitken12 gives a relatively simple demonstration of the theory and includes a series of tables up

This paper gives a geometric illustration of some of the commonest identities in the algebra of the sum of squares. The method is essentially the construction of squares of the various numbers and their means. These pictures provide us with a visual representation of what the identity says. The method is later extended to represent the identities concerning the sum of products by constructing rectangles.

IN MANY forms of analysis of nonnormal variables we seek some transformation of the variate involved in order to make it approximately a normal variable. This enables us to utilize the normal distribution, which has been exhaustively investigated and tabulated, to obtain percentage points, ordinates, and so on for our original variable. If this is not done we may find it necessary to perform much laborious work in the tabulation of the variate itself, and functions of it, perhaps with only limited uses in view for the completed work. A second reason for a transformation is that sometimes the tables of the exact distribution are not available and it behooves us to find a quick and simple test. As an example of the latter consider the standard deviation, s = 2 (x-) 2/n, in a sample of n drawn from a normal population with standard deviation o,. We know that ns2/o-2 is exactly distributed as a x2 distribution but as an approximation we can assume s to be normally distributed with mean a. [1(1/4n) ] and variance c2/2n thus enabling us, at any rate for n larger than about 15, to utilize the normal distribution for a test of significance. Of course, in many cases, where observations fall into a number of groups, it is more important to equalize the variance of these groups than it is to bring the variate into normality. Thus in the analysis of variance, although both items are desirable, it is of more importance, except possibly for a one-way classification, to stabilize the variance otherwise it would never be known if one effect was really significant or not. This present paper examines the properties of a certain class of transformations under the assumption that they are designed to transform the variate, X, into some form of normal distribution. The class of transformat-ions considered is that of