60 THE CONSTRUCTION OF LOG RULES 



Then for any log, 



B.M. = (l-.20 f'^-'-^'' L. 



For a 12-mch log 16 feet long, 



/3. 1416(12 - 

 B.M. = .80' 



^)l6 



48 

 = 92 board feet. 



Method of Deducting Saivdust First. — By the second method, the per cent of 

 waste in saw kerf is first deducted from the entire volume of the log. From the 

 residual volume the amount to be further subtracted for slabs, edging and crook is 

 taken. This is a smaller per cent than by the first method, as shown in Table VII, 

 column 7 since the sawdust used in slabbing is not included, and it is for convenience 

 computed in the form of a plank of width and length equal to the log, and whose 

 thickness is varied to give the required volume of waste. 



Let A equal the width of this plank in inches. This is taken as a constant. 



Then, 



B.M. = ( (1-6)— -AD j-. 



Illustration 



Let 6 = 20 per cent —sawdust allowance, 



A=l .767 inches, the thickness of a plank whose width is equal to D, and 

 length to L — for slabbing allowance. 

 Then for any log. 



B.M..|,8o('-f')-l,767D]^^ 



^12 



For a 12-inch log 16 feet long, 



B.M. = [ . 80( . 7854 X 12=) - 1 . 767 X 12]|f , 

 B.M.= 92 board feet. 



This result shows that for 12-inch logs, after .subtracting 20 per cent from log for 

 sawdust, a plank 1.767 inches by 12 inches gives a deduction from the net volume, 

 equal to method 1 when a collar .75 inch thick is first deducted and 20 per cent for 

 sawdust taken from the remainder. 



The two methods are not absolutely interchangeable. Their relation may be 

 shown by algebraical means. 



Substitute C for (1-6). 



Then C = per cent left after subtracting saw kerf. 



Since D is in inches, and L exerts no influence on the relative values, the areas 

 of the small end of log, left after subtracting total waste, should be equal, and can 

 be e.vpressed in square inches for each formula as : 



CTr(D-a)^ CwTP 



= AD. 



4 4 



Then, 



, C(1.5708aD-.7854a2) 

 A= ;r . 



