XLVIII— XLIX] 



Introduction 



lxxiii 



On p. 91, we have the series for p = 1 when n = 10 for the values m = 5 and 

 m = 10. Taking '6 of the first series and "4 of the second we have : 



The chance that if the two surgeons are of equal skill 4 or more patients will 

 die out of the second surgeon's 7 operations is '058 by interpolation and '047 

 actually. Hence the odds against the occurrence are 16 to 1 by the table and 

 20 to 1 actually. It will be observed that interpolation gives small values at 

 impossible numbers of deaths, but these have to be reckoned in to obtain the 

 total number 100. That all seven patients should die under the second surgeon, 

 if of equal skill, involves odds of 500 to 1 about in the interpolation result, but 

 4000 to 1 about actually. On the Gaussian hypothesis in the original problem the 

 mean - 7 x & = -7 and the S. D. = V7 x ^ x ■& = ' 7 937, and (3-5 - -7)/-7937 = 3-52 

 roughly, or this corresponds to odds of about 4545 to 1 — which are wholly un- 

 reasonable. Thus the Table gives by interpolation odds of approximately the 

 right value, which may serve many useful purposes, for those who are unable to 

 work out the values required from formula (lxxx). At the same time it is clear 

 that a much larger Table with closer values of the quantities involved is desirable. 



Table XLIX (pp. 98—101) 



The Logarithms of Factorials. (Calculated by Julia Bell, published here for 

 the first time.) 



This table was obtained by adding up in succession consecutive logarithms in 

 a table of logarithms to 12 figures. Not until the work was completed did we 

 realise the existence of the splendid table of C. F. Degen*, which was then used 

 to confirm our own results. De Morgan in his Treatise on the Theory of Probabilities 

 of 1837 published an abridgement to six decimals of Degen's Table of Factorials. 

 His values cannot, however, be trusted to the sixth figure of the mantissa. The 



* Tabularum ad faciliorem et breviorem probabilitatis computationem utilum. Havniae, mdcccxxiv. 

 This gives the logarithms of the factorials up to 1200 with 18 figures in the mantissa. 



B. * 



