( 287 ) 
XIII.—On the Tabulation of all Fractions having their Values between 
Two Prescribed Limits. By Epwarp Sane, Esq. 
(Read 21st January 1878.) 
The object sought to be gained by this tabulation may be best seen by con- 
sidering a specific case. When we wish to produce, by help of toothed wheels, 
a certain ratio between the rotations of two axes, we must have that ratio 
represented by integer numbers, and these numbers must contain no prime 
factors exceeding the limit of the number of teeth which can conveniently be 
cut in any wheel. If, for example, we wish to represent the mean motions of 
the planets, we have to express the ratios of their periodic times by decompos- 
able numbers ; and as this, in general, can only be done approximately, we have 
to prescribe some limit of error on either side, and have to seek for all fractions 
between those limits, and then among these to search for those that are 
suitable. 
In my treatise on wheel-teeth a solution of this problem is given, complete 
so far as the discovery of the fractions is concerned, but imperfect in this, that 
it does not place those fractions in the order of their magnitudes. I propose 
now to supply this deficiency by explaining an exceedingly simple process, 
which enables us to make a complete list, arranged in the order of their values, 
of all irreducible fractions whose denominators shall not exceed some specified 
number ; and also to take up and interpolate any portion of the list. This 
process is founded on a general theorem flowing directly from the doctrine of 
continued fractions, but of which the following simple demonstration may be 
given. 
; A C é 
If there be two fractions, — > ay the cross products of whose members differ 
: ; : lee F ; 
by unit, any fraction, as B° intermediate in value between them must be of the 
form — 
B_ pA+qC 
Bo pat+ay 

where p and q are integer numbers, and this fraction is irreducible if p and g 
be prime to each other. 
C 
_If we suppose vy 10 be the greater of the two, so that aC—yA=1, the dif- 
ferences, 
B A ‘aB—6SA C_ B_BC—9B 
Bia igas iat Bat By 
VOL. XXVIII. PART IL. sip 


