I4 INTRODUCTION TO DYNAMICS. [50. 



obtain the result by taking the angle COP =6 as independent 

 variable. We have then 



/^ 



= I 



J 



rcos -r = 2r sn a, 



, sin a 

 whence x=r- 



T-I i_ *.*. 2 r sin a c -, -, .,, 



This can be written x=r =?'-, which agrees with 



2ra s 



the expression found above. 



30. The First Proposition of Pappus and Guldinus. If an arc of 



a plane curve be made to rotate about an axis situated in its 

 plane, it generates a surface of revolution whose surface-area is 

 5=2 IT (yds, where ds is the element of the curve and the axis 

 of rotation is taken as axis of x. On the other hand we have, if 

 s be the length of the generating arc and y the ordinate of its 

 centroid, s-y = \ yds', hence 



5=2 TT sy= 2 iry s, 



i.e. the surface-area of a solid of revolution is obtained by multi- 

 plying the generating arc into the path described by its centroid. 



It is easy to see that this proposition holds even for incom- 

 plete revolutions. When the generating arc cuts the axis, 

 proper regard must be had for signs and sense of rotation. 



31. It follows from symmetry that the centroid of a homo- 

 geneous circular or elliptic area (plate, lamina) is at the geomet- 

 rical centre of figure. Similarly, the centroid of a homogeneous 

 parallelogram is at the intersection of its diagonals. 



In general, if a homogeneous plane figure have two axes of 

 symmetry, the centroid must be at the intersection of these 

 lines since the sum of the moments is zero for each of these 

 lines. 



