138 



G. H. KNIBBS. 



the mean height 1 of the intercepted volume is 



7 _ aT _ V _ „ /A dx dy ,™ 



l -f, x ---f, —j— (25). 



The quantity into which the factor /x is multiplied is obviously the 

 abscissa of the 'centre of gravity' of the right-section of the prism; 

 and hence the volume of any prism with plane ends, is equal to 

 the area of its right section, multiplied by a line perpendicular 

 thereto, passing through its centre of gravity and intercepted by 

 its terminal planes. Since the solid generated by any plane figure 

 rotating about an axis lying wholly in its plane produced, is made 

 up of the infinitesimal or elementary volumes pdO A, in which p 

 is the perpendicular distance between the rotation-axis and the 

 centre of gravity of the plane area, and 6 is the angle of rotation, 

 its volume is 



V=Afpd6 = Apd = l A (26) 



that is to say, the volume of a solid so generated is equal to the 

 area of the right-section multiplied by l , the distance traversed 

 by its centre of gravity: a theorem due to Pappus 2 and redis- 

 covered over one thousand years later by Guldinus. 3 The rotation 

 at a finite rate of the generating plane figure in its own plane 

 about its centre of gravity, while it moves at a finite rate along 

 the curve pdd, is immaterial, and does not modify the result with 

 respect to the generated volume, unless the curve be such that the 

 generator returns on itself, i.e., repasses over any portion of its 

 path : this is evident since the infinitesimal element of the 

 generated solid is always A pdd. 



12. Solids whose longitudinal axes are tortuous curves. — Let 

 the radius of the osculating circle, the circumference of which 

 contains and is determined in position by three consecutive points 



1 That is the height which multiplied by the area of the right section 

 gives the volume. 



2 " Mathematical collections." — Born about 340 A.D. 



3 " Centrobaryca,"— Born 1577, died 1643. 



