216 while M’/C is confined to the limits 0.086-0.123. C may be regarded as the scale factor and M’/C as the shape parameter for the curve. Generally speaking, C’ represents an average over all data points, and its value is rather insensitive to errors in individual points. On the other hand, M’/C is determined by the ratio (cross section at high energy)/(cross section at low energy). Taking cross sections at the two limits of our energy range, the ratio is 0.453 for M’/C = 0.092 and 0.502 for M’/C = 0.123. From these numbers it is evident that M’/C is very sensitive to errors toward either end of the energy range and can be determined only with a much lower relative precision than can C. Conversely, given an accurate value of C, cross sections can be computed quite accurately and extrapolated over a wide energy range in spite of a relatively large uncertainty in M/C. For each gas studied, values of Jf’ and C are derived from a collection of thirty or more data points. A quantitative estimate of the limits of errors is necessanly rather complicated and tedious; the method will be sketched only briefly. The quantities involved are connected bythe relation y = QL[M’x(p) + Cre(p)\(P — Po), (1) where of mec) and is directly proportional to the magnetic field strength in the analyzer P is the pressure of gas in the gas-filled counter. It is convenient to write Eq. (1) in the form n=(1+4)ing 1, p malts p MM C P y/(1 —e= E M+ co | QLOPmin — (2) + [0.124 + X_|Po, where the term in Pp is approximated by the last term. The approximation is permissible because the term is always small and M’/C is never very different from 0.1; é represents the probability that the counting mechanism fails to register a valid ionization act. Suppose that the true values of M’ and C are Mj and Cy and that during the measurements the actual values of p and P differ from those given by the meter readings and ealibrations by Ap and AP, and that « ts not zero, though small. When the observed values are put into a least squares solution for M’ and C, values differing from the true ones by AM’ and AC are ob- tained. We wish to find how AC/Cy and A(M’/C) are related to the e, Ap/p and AP/P. SYSTEMATIC ERRORS yo —In (1 — 9). Systematic errors are those that result from errors of y is the average numberof ionization acts per transit of a primary electron @ is a known numerical constant Z is the path length of the primaryelectron p is the momentum of the primaryelectron (in units 10 + +t+,t + + mum magnitude of Ap/p to be 0.01. Similarly for P, we estimate the error in calibration of the W & T gauge to + _ Pat +o + t+ + + 1 + + ° + 0.5 ““o data point taken at the same p. We estimate the maxi- ¢ 5 + + + o + + + be not greater than 0.05 torr. Counting conditions vary from gas to gas and from pressure to pressure for the same gas. Conditions are tested for each combination of + ot — + - oo | 5 The systematic errors in p may bedifferent for differ- ent values of p, but do not vary from data point to 1 $ calibration for Z, », and P and from nonideal counting conditions fore. The ease of Z is trivial; the uncertainty in the path length is not greater than }5 %; it contributes 0.005 to AC/C and zero to A(M*/C). gas and pressure by observing y at successively higher counter voltages with p near the value for minimum ionization. Conditions are considered satisfactory when the variation with voltage does not exceed thestatistical uncertainty in y; counting is continued until the standard deviation in y, based on counted numbers, is 0.01 (absolute, not relative, deviation). We conclude that ¢ must le between zero and 0.02 except perhaps in a few 10 P/Pmin Fig. 167.—Distribution of data points: +, experiments; ©, calculation of errors. difficult cases. The data points from an actual experiment are distributed generally as indicated by the crosses in Figure 167. The effects of systematic errors must depend upon the area from which the points come, so one can