RELATIVE ACCURACY OF SMALIAN AND HUBER FORMULAE 21 



Provided it has the true form of one of these solids, its volume can be exactly 

 determined by employing the corresponding formula. But the true form of the 

 log may fall anywhere between the fixed points or forms in the series, which are 

 marked successively by paraboloid, cone and neiloid, and in this case the volume 

 even when calculated by the formula which corresponds most nearly to its urue 

 form, will still be in error by the amount of this divergence. This error may be 

 excessive for long logs. 



But by taking advantage of the effect of reducing the proportional height of the 

 frustum, the probable error from this source may be reduced to any desired limit of 

 accuracy. This is done simply by shortening the length of the logs, or by dividing 

 each log into several shorter sections, measured separately. It is then no longer 

 necessary to employ two or more forms arbitrarily according to the variations in 

 the form of the logs, but a single standard geometric form may be chosen, which 

 most nearly resembles the average form of logs, and the same formula; applied to all 

 logs measured. 



The paraboloid comes nearest to answering this requirement, and for this reason 

 the Smalian formula and the Huber formula have been generally adopted for both 

 scientific and practical measurements of cubic volume of logs, to the exclusion of 

 the formula; for cone and neiloid. 



28. Relative Accuracy of the Smalian and the Huber Formulae. 



Logs having the form of a truncated paraboloid are measured with 

 absolute accuracy regardless of their taper by either Ruber's or Smalian's 

 formula. But if the form of the log is more convex and lies between 

 that of the paraboloid and the cylinder, the Smalian formula, measur- 

 ing the two ends, gives too small a result, while the Huber formula will 

 give too large a volume. Nearly all logs lie between the frustum 

 of a paraboloid and the frustum of a cone in form, having slightly 

 convex sides, but not the full form of the paraboloid, so the end area 

 formula (Smalian's) shows an excess, while the middle area measurement 

 (Huber's) gives too small a result. In either of the above cases, 

 the error by Huber's formula Is one-half that of Smalian's and opposite 

 in character. 



Newton's or Prismoidal Forviula. To check the accuracy of measure- 

 ments made on sections of given length and to determine the maximum 

 length of section which will secure the desired degree of accuracy, the 

 prismoidal formula may be applied. This formula is correct for cylinder, 

 paraboloid, cone or neiloid, and consequently for logs of regular form 

 whose volume lies within these extremes. It will not measure accu- 

 rately eccentric or distorted forms resembling none of the above solids. 

 The formula requires the measurement of both ends and the middle 

 section, and is known as Newton's formula. 



When the form of logs resembles more closety the cylinder, cone or 



