462 BELL SYSTEM TECHNICAL JOURNAL 



To apply the above general method in order to obtain for D„ a 

 series more convergent than (21), return to (19-A) and note that 

 when X is small a first approximation for t„ is T„ = \/d„. Then apply 

 (16-A), with y, x, and Oj having the values expressed by (26-A); 

 and g = X, u = t„, U = \/d„, and \l'{u) = {H+d„) tan u. The formulas 

 for the first few successive derivatives of !/-(/<) will be needed, of course. 



Similarly, to obtain for Pi a series more convergent than (22), 

 return to (21-A) and note that when X is small a first approximation 

 for Di is Di = V^ Then apply (16-A), with y, x, and a^ having the 

 values expressed by (26-A); and g = X, u=Du U='V\, and ^(m) = m 

 tan u. 



Graphical Methods for Locating the Tansition Points 



The positions of the transition points D„ (n = l, 2, 3, ... ) on the 

 Z)-scale can be determined also graphically, in several different ways 

 corresponding to several different ways of writing the function 

 (DtanD — X) (DcotP + X) whose zeros are the values of D„. To 

 formulate such graphical methods concisely, let E denote any function 

 of the variable D, so that, geometrically, E is the ordinate corre- 

 sponding to the abscissa D. Six of the various possible graphical 

 methods are then briefly but completely indicated by the following 

 respective statements that the points D„ are the abscissas of the 

 points of intersection of: 



1. The horizontal straight line £ = X with the curves E = D tan D; 



the horizontal straight line£= — Xwith the curves £ = Z) cot D. 



2. The straight line E = D with the curves £ = X col D\ the straight 



line E= —D with the curves £ = X tan D. 



3. The straight line E=D/\ with the cotangent curves £ = cotZ?; 



the straight line £= —P X with the tangent curves £ = tan D. 



4. The hyperbola E = \ D with the tangent curves £ = tanD; the 



hyperbola E=—\D with the cotangent curves £ = cotZ3. 



5. The parabola E = D',\-\ with the curves E = 2Dcot2D. 



6. The curve E = D/'2\-\/2D, compounded of the straight line 



£ = Z)/2X and the hyiH-rbola £=-X 21), with the cotangent 

 curves £ = cot 2/^. 



In nuihiHJs I, 2, .'{, 4. tin- lirsl set of intersections is situated in the 

 odd-numbered segments, the second set in the even numbered seg- 

 ments; each segment of width t/2. 



Besides being susceptible of quantitative service, these graphical 

 methods are useful for qualitative purposes. F"or instance, they show 



