22X Page 74
22X THE ALGEBRA OF PROPORTIONAL BULGES.
(a) The Problem: Recovery of ΔP from ΔD.
It has been pointed out in 22H that the expected letter count of ΔD can be obtained from that of ΔP by means of the equations
(b) Application to Motor Runs.
The problem we are considering here as an application to the question of the expected score on a motor run (R5 23, 32). For we know that
(c) Efforts at Solution.
The problem of solving equations (X1) for P.B. (ΔP) was first attached in R2, 69 where an erroneous connection with 'Fourier Transforms' was suggested. The theoretical aspects of the problem were pursued in R2 p. 87, 104; R3 pp. 24, 28, 32, 34, 37, 38, 48; R5 23 32; and a practical experiment in the solution of ΔP from ΔD is described in R3 pp. 71-3. Finally a relatively simple exposition of the whole subject was given in R5 59. In this chapter we give a still simpler account which contains all the essential ideas, with the introduction of the minimum of new notation. It will be observed that 'Fourier Transforms' are after all the simplest way of treating the problem.
(d) Exposition of the algebra.
Denote an arbitrary teleprinter letter by Θ or Φ. Let ƒ(Θ) be an arbitrary numerical function of teleprinter letters. The Fourier Transform