FORMULAE FOR SOLID CONTENTS OF LOGS 19 



These three soUds form a series of successively diminishing per- 

 centages of the volume of a cylinder of equal basal area and height.^ 

 Each tapers to zero at the tip. But logs are cut with two parallel 

 faces at the two ends. The corresponding solids are the truncated 

 forms of these bodies, termed frustums, as shown in Fig. 3. 



Fig. 3. — Forms of the cylinder, paraboloid, cone and neiloid, and truncated forms 

 or frustums of the last three solids. 



27. Formulae for Solid Contents of Logs. The comparative vol- 

 umes of these four solids are stated by formulae below; when 



B = Area of base, square feet, 



6^ = Area of cross-section, at | height, 



6 = Area of top, 



/i = Height or length, in feet. 



1 Each of these solids is formed by the revolution of a curve about a central axis. 



A true Appolonian paraboloid is derived from that form of a conic section (a 

 symmetrical curve formed by the intersection of a plane with a cone) in which the 

 plane is parallel with the side of the cone. For the conoid formed by the revolution 

 of this curve about its axis, the ratio between a cross section taken at right angles 

 with the axis at any point, and the height above this point to the apex, is constant 



for all points on the axis. This gives a volume equal to — . Logs which taper 



regularly will have straight sides, and resemble a truncated cone. Logs whose taper 

 is most rapid near the butt, diminishing towards the top, will have concave sides and 

 resemble a truncated neiloid. The form and volume of such logs will usually fall 

 somewhere between a neiloid and a cone. Most logs taper more rapidly at the top 

 than at the butt and will have convex sides, and resemble in form a truncated para- 

 boloid — ^their volume usually falls between that of a paraboloid and a cone. Where 

 most of the taper occurs close to the top, the log may exceed the paraboloid in volume, 

 falling between it and the volume of the cylinder. 



