20 



THE MEASUREMENT OF LOGS. CUBIC CONTENTS 



Form 



Volume of 

 perfect solid 



^'olume of FriLstum 



Cylinder 

 Paraboloid 



Cone 

 Neiloid 



Bh 

 Bh 

 2 



Bh 



3 

 Bh 



4 



(B+h), 



Bh 



or (B+h)-. Srnalian's Formula 



b^h. H liber's Formula 



(B+b+Vsly)^ 



3 



{B+ib^+b)-. Newton's Formula 



Newton's formula will also give the volume of the cone, paraboloid 

 and cvlinder. 



The per cent of the volume of the cylinder which is contained in the other three 

 forms, when of equal diameter at base and equal height, is 



Paraboloid 50 per cent 



Cone 33^ per cent 



Neiloid 25 per cent 



But each of these three solids decreases in cross section from base to tip, while that 

 of a cylinder remains the same. The frustum of a cylinder is always a cylinder, 

 while the frustum of a paraboloid, cone or neiloid with equal basal area tends to 

 more nearly resemble a cyUnder as the area of its top section approaches that of 

 its base, which results when the relative height of the frustum is shortened. The 

 per cent of the cubic contents of a cylinder of equul base and height, which is con- 

 tained in these fru.stums increases in the same manner, and the possible limits of 

 variation in form and volume between the cylinder and each of the other three 

 frastums correspondingly dimini-shes. 



E.g., when the height of the frustum is one-fourth that of the perfect soUd, the 

 per cent of cylindrical volume is, for 



Frustum of paraboloid 87 per cent 



Frustum of cone 77 per cent 



Frustum of neiloid Gl per cent 



When the height is one-eighth of a perfect solid, these per cents are: 



Frustum of paraboloid 94 per cent 



Frustum of cone 88 per cent 



Fru.stum of neiloid 77.5 per cent 



A rapidly tapering log forms a truncated section of a relatively shorter completed 

 paraboloid or cone than a log with gradual taper. The greater the height of a com- 

 plete paraboloid with a given basal area, the less it will taper for a given length, as 

 16 feet. \ATiether the taper is rapid or gradual, a log may exactly resemble the 

 frustum of a paraboloid, cone, or neiloid. 



