
HAVING THEIR VALUES BETWEEN TWO PRESCRIBED LIMITS. 295 
679 716 4,9.79.239 





aa = eae eS 
ene pee 242, 364 
aE = eee 242, 398 
Sets won 
me ene en 
a - = ad ee ube 
sa Saay Uaistanag® UNE 
34427 ~ 173.199 
In this way we are able to compute the train of wheels giving a close ap- 
proximation to any proposed ratio, and so could construct a. machine to indi- 
cate say the mean longitudes and anomalies of the planets, with the arguments 
for their perturbations, with a precision commensurate with our knowledge of 
the elements of their motions. | 
The series of fractions thus inserted between two prescribed limits may be 
- ; : - 0 , 
regarded as a small portion of a list extending from unit to on the one side, 
and to = on the other. The filling in of each interval between two fractions 
whose cross-products differ by unit is, in truth, an epitome of the complete 
: ; : See Sue A Ck 
series from zero to infinity. The fraction B inserted between — and y 18 near 
to the former when 7 is greater than unit, and is near to the latter when - is 
almost zero ; that is to say, the complete insertion of all fractions between - 
C i , dy 0 é 
and y Lequires that we assign to - ail values from 9 19 7» represented by in- 
teger numbers prime to each other. In the actual tabulation we are tied down 
to some limit for the magnitude of the numbers ; thus, in the example which 
has been given, B and 8 must be within the limits of our table of divisors, and 
_ the values of p and q are restricted by the condition pA +qB <3 036 000. 
If, after having made a list of fractions represented by numbers under 7, we 
wish to interpolate those represented by numbers up to some much larger num- 
ber N, we may use for p and g the values contained in a more limited list from 
S 0 
or: For the purpose of facilitating such computations, I have constructed 
VOL. XXVIII. PART IL. 4H 
