344 BELL S YSTEM TECH NIC A L JOURNA L 



changing the variable of integration by the substitution 6 = 0/2, and 

 rearranging; thus it is found that 



Q*{R) = ylZ? r ^^P(^^ '^' ^) d<l>. (5.22) 



7r exp L Jo 1 — b cos 



This formula can also be obtained as a particular case of the more general 

 formula (2.24) by setting 6 = 27r in the upper limit of integration, utilizing 

 (2.25), and changing the variable of integration by the substitution 6 = 

 0/2. 



Two partial checks on any general formula for Q{R) = Q{R;b) or for 

 Q*{R) = Q*{R;b) can be applied by setting b — and b — 1, and comparing 

 the resulting particular formulas with those obtained by integrating the 

 formulas for P{R;0) and F'{R;\) obtained in Section 3, namely formulas 

 (3.20) and (3.21) there. It is thus found that 



Q*(R;0) = exp(-R') = ( P{R;0)dR, (5.23) 



Q{R; 1) = 2 |-J= jf^xp -^ dR^=^ [ ^'^^'^ ^^ ^^- ^^'--^^ 



It will be recalled that the quantity between braces in (5.24) is extensively 

 tabulated, and that ^t is sometimes called the 'normal probability integral.' 



Several of the above general formulas for QiR) = p{R'<R) and for 

 Q*{R) = p{R'>R) are closely connected with my 1933 paper." Indeed, 

 formulas (5.11), (5.14), (5.16), (5.19) and (5.22) above are the same as 

 (53-A), (56-A), (52-A), (55-A) and (22-A), respectively, of the unpublished 

 Appendix A to the 1933 paper; and (5.12), (5.13), (5.15), (5.17), (5.18) and 

 (5.20) above were derived in the same connection, although they were not 

 included in the Appendix A. 



Formula (5.22) was employed in the unpublished Appendix A of the 1933 

 paper, being (22-A) there, as a basis for deriving two very different kinds 

 of series type formulas for computing the values of p{R'>R) = Q*{R) 

 underlying the values of pb.t){R'>R) constituting Table I (facing Fig. 8) 

 in that paper. -^ 



2*^ This formula, (5.22), was derived by me in a somewhat different manner in the un- 

 pubHshed Appendix A to my 1933 paper. Later I found that an efjuivalent formula, 

 easily transformable into (5.22), had been given by Bravais as formula (51) in his classical 

 paper ".Analyse mathcmatique sur les probabilites des erreurs de situation d'un point," 

 published in Mcmoires de I'Academie Royale des Sciences do I'lnstitut de FVance, 2nd 

 series, vol. IX, 1846, pp. 255-332. (This is available in the Public Library of New York 

 City, for instance.) 



^^ There the abbreviated symbols p(R' < R) and /)(/?' > R) were used with the same 

 meanings as the complete symbols on the right sides of ecjuations (5.1) and (5.2), respec- 

 tively, of the present paper. 



^^ Each of the two kinds of series type formulas comprised a finite portion of a con- 

 vergent series plus an exact remainder term consisting of a definite integral. In the 



