LAPLACE. 597 



above expression. We will reproduce the substance of his remarks. 

 In Art. 957 we have 



^2 = log r-log^(a+(9) 



= log Y- log !</, (a) + ecp' (a) + | f [a) + .. j 



~ 2 (/)(«) '^"'' 

 for Y= (j) (a), and ^' (a) = 0, by hypothesis. 

 Thus approximately 



Hence if i/ vanishes when a; = and when x=l, we have 

 approximately 



^/(- 





Similarly if we suppose that yz is a maximum when x = a, 

 and that then yz = Y'Z', we have 



j/'^'^^ // d'Y'Z' \' 



Suppose that is a function of 3/, say s = (j) (3/), then 3/2 is 

 a maximum when ?/ is a maximum, so that a =a\ and since 



--— = 0, we find that 

 da 



d'Y'Z' ( . ,_ . ,..,,_) ^^F 



J^ = {<^(r) + rf(r)| 



Hence we have approximately 



P= 



■<^(n 



a/{ 





1031. Laplace discusses on his pages 397 — 401 the following 

 problem. It has been observed during a certain number of years 

 at Paris that more boys than girls are annually baptised : deter- 

 mine the probability that this superiority will hold during a cen- 

 tury. See Art. 897. 



