XIII — XVI] Introduction xxxiii 



Taking froin the column for n = 18 (p. 27) the values for x 2 = 10 and 11, we 

 interpolate P = -891 for ^ 2 =1027 by first differences, and conclude that in 89 out 

 of 100 trials we should get in random sampling a fit as bad or worse than that 

 observed, if the real distribution were Gaussian. Accordingly we say that a 

 Gaussian curve describes excellently the distribution of Bavarian cephalic iudices. 



Tables XIII— XVI (pp. 29—30) 



Auxiliary Tables provided by W. Palin Elderton (Biometrika, Vol. I. pp. 162 — 

 163), useful for calculating values of P for x 2 outside the range of the existing table. 



For such cases we must turn back to the fundamental formulae (xxix) of p. xxxi, 

 and the numerical values of considerable portions of these formulae will be found 

 evaluated in these auxiliary tables. 



Illustration. Find P for n' — 11 and x 2 = 78, ^ 2 = 39, hence by formula 

 we have 



= e- 39 (40 + 760-5 + 9886 5 + 96393375) 

 = e- 39 x 107080-375, 

 where the powers of 39 are taken out of Table XXVII (p. 38). 

 Hence using Table XIII, 



logP= 170625,1520 + 50297,0988 = 12-0922,2508, 

 which gives us P = T23659/10 12 . 



As a rule we can select n to be odd, but, if it is necessarily even, there is more 

 trouble, not in the determination of the series, but in the evaluation of the 

 integral 



e' hy? d X . 



-vi£ 



A table of the values of F=^I for % = ^ fco 500 has been given as Table IV 

 (p. 11). This gives X 2 = 25 to 250000 but the intervals are large. 

 If greater accuracy be required then Schlomilch's formula* 



1 



v ttJ x * v tt* 6 y x \ x ^ 



+ 



! +2) X 2 (X 2 + 2 )(X 2 + 4) 

 5 9 



X 2 (X 2 + 2) (x 2 + 4) (x* + 6) x* it + 2) (X 2 + 4) ( X 2 + 6) <tf + 8) 



129 \ 



X 2 (X 2 + 2)(x 2 + 4)(x 2 +6)(x 2 + 8)(x 2 + 10) + "-[ (***) 



must be used. 



Here a/ — X e ~ w '" be found in Table XIII, and the series converges fairly 



rapidly. 



* Compendium der h'dheren Analysis, Bd. n. S. 270, Braunschweig, 1879. 



