478 BELL SYSTEM TECHNICAL JOURNAL 



If we write Ai, A2, etc. for the probable errors involved in the 

 measurements of Di, Do, etc. we find on solving for amplitudes and 

 phases, and compounding errors ^ that 



fli — 21) ^ °^^ + fli ) '", 



^2 COS 0:2 = — . y-. ± d(l + a2*^ cos-^ 0:2) '", 



where 



8 = (Ai2 + A22 + A3^ + A^yyiDo. 



Substituting into these formulas the values of D and A contained in 

 Table II, we find 



ai = 0.0169 ± .0080, 



tan ai = - 0.50 ± .45 



(136° < ai < 177°), 



aa cos «2 = - 0.0018 ± .0040. 



The last of these quantities includes the amplitude of the polarization 

 effect as one of its components. To make this explicit we may restrict 

 a2 and 0:2 to represent the amplitude and phase angle of variations of 

 twice the fundamental frequency due to mechanical imperfections only, 

 and use p to represent the amplitude of the polarization effect. We 

 may then write, since the phase angle associated with p is zero, 



p + a2 cos a2 = - 0.0018 ± .0040, 



and from this we wish to infer that p is itself a small quantity, the 

 same in order of magnitude as (p + a2 cos 0:2). 



It may be urged, of course, that nothing in regard to the value of p 

 is to be inferred from the value of (p + 02 cos 0:2), and this in a strictly 

 mathematical sense is true enough; the individual terms may both be 

 large, and the small value of their sum may be entirely fortuitous. 

 While one must recognize this as a possibility, he must recognize also 

 that the likelihood of the occurrence of chance compensations of such 

 perfection in the case not only of this beam, but in the cases of the 

 others as well, is extremely small. The values found for (p + ao cos 012) 

 for all five beams have been set down in Table III. It will be seen that, 



■^ In calculating the probable errors of these functions we have disregarded the 

 small differences in precision involved in the measurements of the various deflections. 



