288 MR EDWARD SANG ON THE TABULATION OF ALL FRACTIONS 
must both be positive, and we may write aB—BA=g, BC—yB=p, p and q 
being necessarily integer and positive. On eliminating first 6 and then B from 
these, we obtain— 
(aC—yA)B=pA+ qC; (aC—yA)B=pat gy, 
that is— 
B=pA+qC; B=patqy. 
Hence, if we have two fractions, as > and s we may obtain others inter- 
mediate between them by inserting in the formula | 
5p + 8q 
Tp+11q 

any positive values for p and q, observing, however, that if p and g have a 
common divisor, the resulting fraction is reducible. 
The fraction 7 however, so inserted between 5 and - does not necessarily 
give, with either of these, cross products differing by unit, for we have— 
aB—BA=q(aC—yA)=¢ 
BC—yB=p(aC—yA) =p, 
and thus the same law of interpolation does not hold good; but if p and q be 
each taken as unit, the cross products on each side differ by unit. 
and 
From this we see that the lowest fraction intermediate in value between : 
Or aapaelics ; ‘ 6 13, 8 
and 7; is zg’ and that each interval of the progression , 7>]g9qz» may be 
use io 2d 
. 5 
filled up in the same way, so as to give 7’ 95’ 18’ 99° i’ 
If, then, we wish to insert between the limits : and = all fractions whose 
denominators shall not exceed, say 50, we have only to continue this process 
until the sum of the contiguous numbers do not exceed 50. Our list is then— 
5 83 .28° “258 WA oa) Dl 29S 
7 46° 89 (oo eo re ty ear20 20 otal 
As an interesting application of this method, I shall propose to compute 
those trains of wheels which shall convert mean solar into mean sidereal time 
so accurately as not to err by one second in the year. 
For this we observe that, according to the computations by BEssEL, the 
equinoctial year consists of 36524 221-7 solar, or of 36 624 221-7 sidereal 
seconds, and that, therefore, these numbers represent the ratio to be produced 

