THE EXTRACTION OF THE JUICE BY MILLS 

 Whence from (i) and (2) it follows that 



205 



or 



w r 



log ^ = - u 



R 



If, for sunplicit}', m be \\Titten for—, and if R be put equal to (i), this 



equation reduces to log r = m.u 



or y = e"*" . . . . . . (3) 



The equation (3) is none other than that of the logarithmic spiral where 

 e is the base of the natural system. 



This curve has the property that the radius vector always makes with the 



tangent a constant angle ; thus the angle 

 a is constant. Now (see Fig. 104), 



Fig. 105 



m = — = tan s 



w 



and g = go"^ - a 



then m = cot a = constant. 

 Draw ON perpendicular to OA, and AN perpendicular to T. 

 Then ON = r cot a = rjn. 



In addition NA 



Equation (3) gives the path 



-y/ I + m^ sin a 



which the point p describes as a logarithmic spiral ; for sugar miUs this curve 

 is of definite length. 



The path, then, which the point p and also the crushed cane describes is 

 a part of a logarithmic spiral ; in order to obtain this path for sugar mills the 

 velocities w and c must be known. The velocity u.-, which is perpendicular 

 to the radius vector, is always equal to T, the velocity with which the bagasse 

 leaves the first cylinder pair. The velocity c is to be determined experi- 

 mentally, and depends on the elasticity of the crushed cane, and that the 

 cylinder 3 must easily carry forward the bagasse. Before determining 

 empirically the values of c and of the angle a, we wiH look first at the following 

 considerations : — 



In Fig. 105, S is the opening between the cj-linders i and 2, and d is the 

 thickness of the crushed cane, and when the cane is not elastic d is equal to S : 

 in this case the velocity c can be put equal to 0, for there exists absolutely no. 

 reason why the crushed cane should proceed with a velocity c hdng in the 



