LAPLACE. 523 



Therefore approximately 



d<t> V6 



dt ~^ls{m'-l)}' 



The limits of ^ will be and x . Hence approximately 

 2 r^ /sin 7?2(f)\* ,^ 2 w'' ^/6 r -t^ j. 



m' V6 (2w + 1)' \/3 



Laplace next considers the value of the integral with respect 



to <h between the limits — and — , and then the value between 

 ^ m m 



the limits — • and ^ — , and so on ; he shews that when 5 is a very 

 771 m 



large number these definite integrals diminish rapidly, and may 



be neglected in comparison with the value obtained for the limits 



and — . This result depends on the fact that the successive 

 m 



Sm 'ITZCD 



numerical maxima values of — -. — r^ diminish rapidly ; as we shall 



sm ^ 



now shew. At a numerical maximum we have 



sin vi(f) m cos m(f) m m 



sin ^ cos ^ cos V(l + ^^'^ tan^ </>) VCcos'^ ^ + m"^ sin^ 0) ' 



this is less than ^ — r , that is less than . . . — , and therefore 



sm (p sm (p 9 



a for^tiori less than — — , that is less than ^ — r . 

 •^ 2 <j) 2 incp 



Hence at the second maximum —. — j- is less than - -— , 



sm 2 5 



that is less than ^ , and therefore the ratio of the second nume- 



o 



