421 



argument of probabilify, because it is the argument with 

 which Encke's Table should be entered, in order to obtain 

 from that Table the value of the probability p. He shows 

 how to improve several of Laplace's approximate expressions 

 for the argument t, and uses in many such questions a trans- 

 formation of a certain double definite integral, of the form. 



— \ c?r \ due~^^ v co&{2s'ru\) 



TT J J„ 



in which 





U = 1 + ai M^ + 02 M + • • • 



v = 1+j3im' + /32«'4-..- 

 while vi, V2, • . . depend on oi, . . . /3i, . . . and on r ; thus 



v\ = ^ai — /3ir^. 

 The function 6 has the same form as before, so that if, for 

 sufficiently large values of the quantity * (which represents, 

 in many questions, the number of observations or events to 

 be combined), a probability^ can be expressed, exactly or 

 nearly, by the foregoing double definite integral, then the 

 argument t, of this probability p, will be expressed nearly by 

 the formula, 



t = r(l + viS-'+V2S-''). 



Numerical examples were given, in which the approxi- 

 mations thus obtained appeared to be very close. For in- 

 stance, if a common die (supposed to be perfectly fair) be 

 thrown six times, the probability that the sum of the six 

 numbers which turn up in these six throws shall not be less 

 than 18, nor more than 24, is represented rigorously by 

 the integral 



2f*"~ , sinTa; /sin6^\^ , , ,. . .,..o 

 « = -\ dx— l^-^ — , or by the fraction fllM ; 



while the approximate formula deduced by the foregoing 

 method gives 27449 for the numerator of this fraction, or for 

 the product 6^p ; the error of the resulting probability being 

 therefore in this case only 6~^. The advantage of the method 



