CANE SUGAR. 



Substituting this value of C 2 it follows that 



u = log (R -\- ct) log R 



or u = ~ log R + et (2) 



Whence from (1) and (2) it follows that 



10 r . 



r c 

 or log R = w u 



If, for simplicity, m be written for , and if R be put equal to (1), this 



equation reduces to log r = m.u 



or r = 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. 116), 



c 



m = =. tan a 

 w y 



and g = 90 - a 



then m = cot a =. constant. 



Draw ON perpendicular to OA, and .4 A 7 " perpendicular to T. 

 Then ON = r cot a = rm. 



T 



In addition NA r r~i i ?- Equation (3) gives the path 



\/ 1 -|~ m 2 sin a \ / o 



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

 is of definite length. 



186 



