

290 MR EDWARD SANG ON THE TABULATION OF ALL FRACTIONS 
whose values are between-the prescribed limits, we obtain 173 cases, of which, 
however, only eight can be resolved into products of primes less than 240, 
which may be taken as the limit of the number of teeth for a clock-wheel. 
These cases are shown in the following list :— 








' Fraction. Year. Annual Error. 
moni sae | ede | 4102 
ae sess | 298 |. + 2 
SE ee £89 
sooo = agaor | 9264 | + 
sod = Troe | bie eee ee oa 
ree ee ee ao 
may = ieiisy «| 290% | - 2 
34333  13.19.139 DMRS wae 

34427 «173.199 

The fifth case in this list gives a train of low-numbered wheels, viz., 
37 39 ot 
34° 44° 59’ 
which gives an annual gain by the sidereal index of only ‘42 of a decimal 
second ; that is, of ‘36 of a common second ; that is to say, an error of about 1 
second in three years. 
By inserting fractions in the intervals of these 173, we obtain others with 
larger numbers, and among these we may find new cases available for our pur- 
pose. But we have already obtained a low-numbered train with an error in 
defect of only 155°3 in the length of the year; wherefore it would be a waste 
of labour to compute the fractions outside of that degree of accuracy ; where- 
fore we may now restrict our inquiries to ratios between the limits 
365'240 664 : 366°240 664 and 
: 865°243 770 : 366°243 770 ; 
ér, what is the same thing, we have to insert fractions in the intervals of the 
88023 73414 
series from -55; 38064 '° 73615 already found, there being 71 of these intervals. Of 

