522 LAPLACE. 



and between each of these values it will be found that the ex- 

 pression is numerically a maximum, and it is also a maximum when 

 (j) = 0. Thus we may calculate by Art. 957 the value of the integral 



'sm 7nd)\^ . . . . TT 



— : — ~ d6 when the limits are consecutive multiples of — . 

 ^ sni (p J ^ m 



sm Tyicri 

 The equation which determines the maxima values of — ^ — ,- 



sm<p 



is 



m, cos mcf) sin (/> — cos <j) sin mcf) 



!• 



sin^ (j) 



0. 



It will be found that this is satisfied when <^ = ; the situation 

 of the other values of cf) will be more easily discovered by putting 

 the equation in the form 



tan m(j) — m tan (^ = : 

 now we see that the next solution will lie between m(b = -— and 



»">73" 7 77" 077" 



m^— — , and then the next between m(p = -r- and m<^ = -^, 

 and so on. 



We proceed then to find 



''sin mcf)\ 



IT 



'711 



sm 



JJ 



d(f>. 



The maximum value of the function which is to be integrated 

 occurs when 4> = 0, and is therefore m* ; assume 



— 77i*e~^', 



/sin m(f)^ * 

 V sin (/) , 



j = m*e~''; 



take logarithms, thus we obtain 



