558 CHAPTER XXVII 



finally purified material whence sugar is removed as crystals. As long as 

 the principle of the formula is understood, the basis of reference is a matter 

 of indifference. 



The general control formula discussed in detail above may be used as a 

 starting point to deduce other formulae of use in control. These, which 

 follow from simple algebraic transpositions, are collected below, and from 

 them passage by means of constant multipliers ma}' be made to commercial 

 standards of reference, such as " gallons of molasses per bag." 



Let s denote the purity of the final product : raw sugar, or refined sugar, 

 in which case s = i. 



Let j denote the purity of the initial material : — syrup in a raw sugar 

 house and raw sugar in a refiner}^ 



Let m denote the purity of the by-product : — molasses in a raw sugar 

 house and " barrel syrup " in a refinery. 



Then : — 



1. s [j — m) Ij (s — m) = sucrose in product per i sucrose in initial 

 material. 



2. (y — m) jj (i — m) = product = sucrose in product = solids in 

 product per i sucrose in initial material, when referred to refined sugar 

 as product. 



3. m (s — j) I j {s — ni) = sucrose in by-product per i sucrose in initial 

 material. 



4. w (i — j) Ij (i — m) = sucrose in by-product per i sucrose in initial 

 material, when referred to refined sugar as product. 



5. j — m/s — m — solids in product per i solids in initial material, 



6. j — ni/i — m — product = solids in product per i solids in initial 

 material, when referred to refined sugar as product. 



7. s —j /s — m — solids in by-product per i solids in initial material. 



8. I — y/i — m = solids in b37-product per i solids in initial material, 

 when referred to refined sugar as product. 



9. s — j /j — m — solids in by-product per i solids in product. 



10. I — j Ij — m = solids in by-product per i sucrose in product, per 

 I solids in product, per i product, when referred to refined sugar as product. 



11. m {s ~ j) js {j — m) = sucrose in b^^-product per i sucrose in product. 



12. w (i — j) Ij — w = sucrose in by-product per i sucrose in product, 

 per I solids in product, per i product when referred to refined sugar as 

 product. 



Again, if ^ = sucrose in initial material, especially raw sugar, then 

 13- P ij — ^) IJ (^ ~ ^0 ~ sucrose in product = solids in product = 

 product per i of initial material, referred to refined sugar as sole product. 



14. Non-sugar in raw sugar /non-sugar in barrel syrup = barrel syrup 

 per I raw sugar, where refined sugar is the sole product. 



15. If Sj and Sg be purities of the product, then 



sucrose in raw sugar of Sj purity _ s^ (sg — m) _ s^ — m _ 



sucrose in raw sugar of Sg purity s^ (s^ — m) s^ (i — m) 

 i.e., with reference to pure sugar. 



These formulae may be used to solve man\^ problems, some examples 

 being appended. 



I. What are the comparative weights of raw sugar of composition 

 (a) 96-0 per cent, sucrose, 96-3 gravity purity, and (b) 97-0 per cent, sucrose, 



