86 BELL SYSTEM TECHNICAL JOURNAL 



on Fig. 3B, and the values formerly entered at 90 (points A, B, etc.) 

 entered at 813^2 (points a, b, etc.). With these points and those at 

 100 as a guide, new values for 90 allowance have been estimated 

 (points A', B', etc.). The shifting of a point up or down on Fig. 3B 

 results in shifting the corresponding points of the other family (Fig. 

 3A) the same distance in the same direction resulting in the dotted 

 curve. There is much more smoothing necessary to make Figs. 3A 

 and 3B satisfactory and reasonable, but by proceeding in the manner 

 just described, taking into account the appearance of the curves, the 

 logic of the situation and the original data, a smooth and consistent 

 family of curves can finally be evolved. 



Another excellent method of smoothing involves the use of a three 

 dimensional figure. Just as a plane surface gives a complete repre- 

 sentation of two variables and a partial representation (by coding) 

 of a third, so a three dimensional system can be used to give a com- 

 plete representation of three variables, and a partial representation of 

 a fourth. It also aids greatly in smoothing simultaneously. A 

 device for three dimensional representations consists of a plane sur- 

 face marked off with rectilinear coordinates and having at frequent 

 intervals holes into which pegs can be set. The pegs also have co- 

 ordinate markings. The values of two variables, then determine the 

 point at which the peg is set and the value of the third determines 

 a distance along the peg. The point is marked by a small rubber ring 

 which fits around the peg. The values of a fourth variable may be 

 coded by using rings of different colors. When the device is used for 

 smoothing curves involving only three variables, the data points may 

 be indicated in one color and the smoothed values in another. The 

 data points, remaining constant as the smoothed curves are shifted, 

 form a continuous check on the divergence of the smoothed curves 

 from the data. This is an automatic process of cross-sectioning. 

 When the position of a point is changed, the effects of the change on 

 the various relationships are seen by studying different aspects of the 

 setup. This device gives the best results with discontinuous variables 

 (such as message allowances, rates, etc.) as the pegs can then be set 

 in at regular intervals without resorting and regrouping the data. 

 It is also especially valuable when one of the variables is a complex 

 factor (such as distribution of development among more than two 

 classes of service) which cannot easily be represented by one curve. 



Fig. 3C illustrates such a setup with the revised curves of 

 Figs. 3A and 3B. The independent variables, message allow- 

 ance and rank of service are represented by the rectilinear co- 



