HAVING THEIR VALUES BETWEEN TWO PRESCRIBED LIMITS. 291 
fractions whose denominators are below 200 000, we find 284; of which, there- 
fore, 212 are new, or between 100 000 and 200 000: and among these we find 
only three with prime factors less than 240. These are— 

Fraction. 

119799  3.9.9.17.29 
OAT | Ass 
S188), BOIS 
178 360 — 8.7.7.7.13 
183 717 3.3.137. 149 
184 220°” 4,.5.61.151 
133 679 _ 7.13.13.113 
134 045 — 5.17.19.83 


Year. 
‘240 854 
‘242 300 
"242 545 
243 169 

Annual Error. 
+ °35 
— ‘02 
— 09 
— ‘26 
among which is written one fraction having the prime factor 281, but notice- 
able on account of the very ciose approximation. 
Restricting now our inquiries to fractions representing the year as between 
365°241 890 and 365:°242 545, that is, to a limit of the error of the clock of one 
second in eleven years ; and, at the same time extending the range of the de- 
nominators to 400 000, we obtain a new series of approximations. 
We have already 60 fractions within these limits; on interpolating we get 
177 new ones, among which three only have their members decomposable into 
prime factors sufficiently small ; these are— 
Fraction. 
266992 — 1611.37.41 
cy i 9.151.197 



271375 _ _125.13.167 
272118 ~ 23.7.11.19.31 
256035 _ 3.5.13.13.101 
256736 | 32.71.1138 
Year. 
242 134 
242 261 
‘242 510 
Annual Error. 
+ 023 
— 012 
— ‘080 
one of which errs by only the 80th part of a decimal second in the year. Hence 
in our further search we may narrow the limits of the error to between 
365°242 171 and 365-242 261 for the length of the year, that is, to between the 
fractions 


On continuing this interpolation to all denominators less than one million, 
VOL, XXVIII, PART II. 
46 
