CANNING SARDINES 165 
0.046 unit of sardine oil remained in the frying oil for each unit of large, fat 
fish eooked under conditions similar to those used in cooking the lean fish. It 
is possible, by using the figure (0.035) for the amount of oil that cooks out of 
the fish in the bath and the amount of oil removed from the bath, to make 
calculations that show at least how rapidly the sardine-oil content of frying oil 
must increase when large, fat sardines are fried, the oil content of the bath 
remaining constant. Use of a value larger than 0.035 would give figures showing 
a more rapid increase. 
The average amount of sardines fried at one time in the second run of frying 
experiments was about 4 pounds. Calculations were made, therefore, on 
4-pound units. It was also assumed that the quantity of oil in the bath remained 
at 12 pounds. The amount of oil that cooked out of each 4 pounds of sardines 
was taken as 0.035 pound 4=0.140 pound. The same amount of mixed oil 
Was assumed to be mechanically carried out of the bath by each 4 pounds of 
sardines fried. After cooking the first unit there would be 12.14 pounds of oil 
in the bath. Mixing of the oils would be quite complete, due to the action of 
steam coming from the fish and bubbling constantly through the oils. On 
lifting the fish out of the oil a quantity of the mixed oils would be carried out of 
the bath. Part of this would return during the draining period and the rest 
(0.140 pound) would be permanently removed. ‘The fish oil left in the bath 
would be 12.00.140/12.140. The fraction (12.0/12.140) of the amount of 
fish oil in the bath would remain after each unit was removed. In general— 
Let a=pounds of fish oil that cook out of the fish for each unit cooked. 
b=fraction of fish oil remaining in the fry-bath oil left behind after 
~ each unit is cooked and removed. 
Xn=pounds of fish oil in the bath after n units of fish have been cooked 
and removed. 
x — 0a; 
X2=b (ba+a)=a (b?+5). 
=p ib (ba+a)+al=a (b?+62--b). 
X,=a (Gt-- 0° 4- b?4-b). 
et Ot eto one 0 (4) b= (81) [or X, =X, 14-07 
The last equation shows that each succeeding calculation can be made by 
adding ab" to X,;. In this case ab" is only bXab™". Each succeeding unit 
ean therefore be found by adding log. b to log. ab™™, then finding the number 
that corresponds to this sum and adding this number to the preceding value 
of X,_; to get the new value of X,. If a number of these calculations are 
made, it will be seen that log. ab=—log. ab®~“ is constant for the different values 
of n. This difference, when successively added to the logarithm of ab, gives 
the succeeding logarithm of ab™. The value of this logarithm is then added to 
the value of X,_; to get X,. In finding the value of X, the calculations are 
simplified by multiplying the difference in the logarithms mentioned above 
by n-1 and adding it to the first logarithm. The difference is then successively 
subtracted (algebraically) from this new logarithm. This gives a list of 
logarithms. The numbers that correspond to these logarithms are then written 
down and added to get the value of X,. This mode of attack, used with the aid 
of a tabulating adding machine, greatly simplifies calculations. The equation 
and method of calculation developed is applicable to other problems where the 
amount of oil in the bath remains constant. 
The values used in this set of calculations are given below. The results of the 
calculations are listed in Table 20. 

n= 200 
a=0.140 (log. a=9.1461—10) 
12.0 
[j= 12.140 (log. b=9.9950—10) 
log. ab®—log. ab™ i=0.0050 
Another set of calculations was made, assuming the same conditions to exist, 
except that 1-pound units were taken. Results are shown in Table 21. 
Other calculations along similar lines were made, which show the effect on 
the sardine-oil content of fry-bath oil when the oil content of the bath increases. 
Two such calculations were made—one to show how the increase takes place if 
