LAPLACE. 



56o 



Up to this point the investigation has been exact: we shall 

 now proceed to approximate. Suppose 5 to be a very large num- 

 ber; then Y is the product of a very large number of factors, each 

 of which is less than unity except when x =0. We may infer that 

 Fwill always be small except when x is very small; and we shall 

 find an a^Dproximate value of Y on the supposition that x is small. 



Let 



ra 



I zfi {z) dz = Ici, 



J b 



z'fi {z) dz = ki', 



J b 

 fa 



z^fi (z) dz = ki", 



J b 



z\fi {z) dz = kl", 



Then we shall have in converging series 



.4..4 7 '" 



pi cos r^ = 1 — 



^^ikl x^^ik, 



+ 



7 ^V-^/' 

 pi sm Vi = X'^iki r^ f- 



Let 



2 



\3 



{k- — kl) = hi ; then we obtain 



Hence 

 therefore 



pi=l —x^^m^- , 



Ti — X'^^^i-V 



log /?i = - a^V V + ; 



p. = e"^V/u2 approximately. 



Let k'- stand for ^<yihi, and I for ^Yi^'i, each summation extend- 

 ing for the values of i from 1 to 5 inclusive. Then approximately 



Thus (2) becomes 



dx 



2 r o o . dx 



P = — e"""^^ cos ilx — cx) sin rjx — 

 7rj„ ^ ^ ' x 



(3). 



The approximate values which have been given for Y and 2/ 

 can only be considered to be near the truth when x is very small ; 



