HAVING THEIR VALUES BETWEEN TWO PRESCRIBED LIMITS. 289 
by the wheel-work. Also, the coincidence of the beats of the solar and sidereal 
clock occurs at intervals of 365 solar or 366 sidereal seconds, wherefore an error 
of that much in the estimated length of the year would produce a discrepancy of 
one second: between the two clocks. The limits of the ratio become, then, 
36 524 586-9 36 523 8565 
on the one hand 36 6945869 ’ and on the other 36 6238568’ 
and our business is to make a list of all those fractions whose values are 
between these limits. Moreover, since the members of these fractions have to 
be resolved into their factors, and since BurcKHARDT’s table is the only one 
available for this resolution, we have the further restriction that the members 
of the fractions shall not exceed three million. 
On converting these into continual fractions, we find the quotients 1, 365, 
4,14, &., and 1, 365, 4, 5, &c., respectively, giving the successive approxima- 
tions 
365 1461 7670. 
and ——-» —— 
365 1461 20819 
5] ee ae 
366. 1465 7691" 
366 1465 20876" 
wherefore every fraction between the prescribed limits must be of the form 
365p+1461q¢ 
366p+1465q 
in which g exceeds 5p, and is less than 14p; hence, assuming p=1, and 
making qg successively 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, we obtain the series 
Soyo) 9181 10592 12053 138514 14975 16436 17897 19358 920819.. 
7691’ 9156’ 10621’ 12086’ 13551 15016’ 16481’ 17946’ T9411’ 30876’ 12 
which the cross products of every adjoining pair differ by unit; so that no 
fraction in lower terms can have its value intermediate between any two. 



If we agree to use only denominators under 21,000, we may insert Pr 
9723 
between the first and second, ie between the second and third. 
On examination, by help of BurcxHarpt’s table of divisors, we find that not 
one of these twelve ratios can be resolved into others admissible in clock-work. 
. 10892 : 2.2.2.2.2.331 
tay 10592 2.2.2.2.2.331 . 
For example, the fraction ee be written CRT Cute but 331 is an 
inconveniently large number for the teeth of a clock-wheel. We must, there- 
fore, go to fractions with higher denominators. 
The most convenient way of arranging the work is to write each fraction on 
a separate card, placing the cards in their proper order: to interpolate between 
two, we write the sums of the two members on a new card, which is then 
placed between the others, and so is ready for other interpolations. 
On computing in this way all fractions with members of five places, and 
