DIFFERENTIAL AND INTEGRAL CALCULUS. 43 



6 being an angle less than ^TT, since we take = 0, when 

 e = 0. Similarly 



*-log(l-sin0siDa:) , 



/: 



smu? 



I" 71 " 1 . /I + sin# $\nx\ , 

 whence I . - log I _ . . dx = 2-7T0 

 7 smx \i sine/ smxj 



j i [* 1 i /I + SH10 sinicN , 



and also -. log :\dx Trd. 



J sma; \l - sm# smxj 



/' ir , /l + sin(9 sinic\ 7 

 and -: lag ^ 7r . dx TT 0. 



J sin a; \1 sin^ sinoy 



T-TT ] 



Also I - log ( 1 - sin 2 9 sin 2 a?) dx - 

 j o * 



20 

 a; 



riTT /7>. 



= 2 -Iog(l-sin'^sin 2 



J 811105 



/TT ft 

 -r log (1 - sin 2 sin >2 j; 

 . BIUU5 



, 



log (.... } = - 



EXAMPLES OF THE USE OF THIS METHOD. 



fiw 



1. Determine the value of I tan" 1 [m V(l tan a ic)) dx. 



J t 



2. If a string just surround a closed oval curve, and 

 another curve be formed by unwinding this string, be- 

 ginning at a point P, and unwinding the whole, the whole 

 arc of this involute will be a maximum or minimum when 

 Pis such a point that the perimeter of the circle of cur- 

 vature there is equal to the perimeter of the oval, a maxi- 

 mum if unwound in the direction in which the curvature 

 decreases from P, and a minimum for the opposite direction. 



3. A string of given length is fixed to a curve at (9, 

 and laid along the arc OP of the curve, it is then unwound 

 and bound on to the curve on the other side of along 

 the arc OP'. Prove that the arc traced out by the end 



