540 Mr. stokes, ON THE CRITICAL VALUES OF 



If we replace the product of the sine and cosine in this expression by the sum of two sines, 



by means of the ordinary formula, and omit unnecessary constants, we shall have for the series 



to consider 



] . ^ ^ 



2 - sin jjs; , (12). 



n 



Let now 2< = sin s; + * sin 2xr ... + - sin wsr, (13), 



■^ n 



du sin(ra+i)sr 



then -;— = cosxr + cos 2 « ... + cos nz = : — r^^ — - A : 



dz 2 sin ia: 



2 



and since u vanishes with sr, in which case : — t^~-~ is finite, we shall have, supposing z to lie 



between -2 7r and + 2 tt, so that the quantity under the integral sign does not become infinite 

 within the limits of integration, 



, /•- sin (n + 1) sr , ST , ^ 



-^ J^ sin 1 s; 2 



and we have to find whether the integral contained in this equation approaches a finite limit as n 

 increases beyond all limit, and if so what that limit is. Since u changes sign with z, we need not 

 consider the negative values of z. 



First suppose the superior limit z to lie between and 2 tt ; and to simplify the integral write 

 2 z for z, w for 2 w + 1, so that the superior limit of the new integral lies between and tt ; then 



r'&mnz , r'smnz z , I'imnz „ ^ , 



the integral = / —. dz = / : dz = / {l+Rz)d%, 



° Jg Sin z Jo ^ sm z Jq z 



z ^ Sin z 

 where R = , a quantity which does not become infinite within the limits of integration. 



;r sin s- 



Hence, as is known, the limit of f^ sin w^ . Rdz when n increases beyond all limit is zero. Hence, 



if / ))e the limit of the integral, 



,. . „ /•''sin war , ,. . „ r"' sin J' .^ 

 / = limit of / dz = hmit of / -t^ dT- 



Jo ^ -^0 ^ ^ 



Now, z being given, the limit of wsr is eo , and therefore 



Secondly, suppose z in (14) to be equal to 0. Then it follows directly from this equation, or 

 in fact at once from (13), that u = 0, and consequently the limit of m = 0. 



The value of u in all other cases, if required, may be at once obtained from the consideration 

 that the values of ti recur when z is increased or diminished by 2 tt. 



Hence, the series (12) is in all cases convergent, and has for its sum when z = 0, and ^ (tt - z) 

 when z lies between and 2 tt. 



Now, if in the theorem of Article 4. we write z for ,r, and put a = tt, f(z) = 1 (■n- - z), we find, 

 for values of z lying between and tt, and for sr = tt, 



limit of 2 — g" sin jiz = i (tt — «■) ; 



