HAVING THEIR VALUES BETWEEN TWO PRESCRIBED LIMITS. 297 
adjacent fractions, we have— 
p29 20 
— —»,» — 
iy 401. 93" 
where the difference 133 still subsists. Another step gives us— 
9 38 29 49 20 | 
Teens 40 (62, ,oax 
where we observe that the second fraction may be simplified by the divisor 19, 
the fourth fraction by the divisor 7, thus giving— 
beat 2 29 lng 20 
en 40 9? 93" 
In the first and second intervals the difference is 7, in the third and fourth 
19. The next operation gives— 
So A a 81 29 86 ar 20 
EK = ares ae) 
p20 On ta AO 401), 9) 05082 D3 
which are all in their lowest terms ; the differences of the cross products remain- 
ing 7 and 19 as before. In the same way we may continue until all the differ- 
ences become unit. 
Instead of proceeding by simple addition, that is, by using p=1, g=1, we 
may determine p and q directly, so as to give a common Givisor 7 or 19. Thus, 
taking the first pair of the above series, we may resolve the indeterminate 
equation 99+11g=7n, or 1p+2q=7n, giving ap qg=3, whence the inter- 
6 
42 : shen) 6 
mediate fraction = The list then is ie ie = » &e. 
Though this method appear to be more direct, it is in reality much more 
laborious than that which we used at first. 
Table of all Ratios represented by Numbers up to 20. 

050 000 1:20 | 20-000 000 150 000 3:20 6666 667 
052 632 1:19 | 19-000 000 153 846 2:13 6:500 000 
055 556 1:18 | 18:000 000 "157 895 Beal, 6°333 333 
058 824 Peres | L7-000'000 "166 667 1:6 6-000 000 
062 500 1:16 | 16-000 000 ‘176 471 a: 17 5°666 667 
066 667 1:15 | 15-000 000 181 818 POA 5°500 000 
"071 429 1:14 | 14:000 000 ‘187 500 3:16 5°333 333 
076 923 1:13 | 13-000 000 ‘200 000 LES) 5:000 000 
083 333 1:12 | 12-000 000 ‘210 526 4:19 4°750 000 
090 909 1:11 | 11-000 000 ‘214 286 3:14 4666 667 
"100 000 1:10 | 10-000 000 "222 222 Za 4:500 000 
105 263 2:19 9°500 000 ‘230 769 3:13 4333 333 
“laa 1) 9-000 000 ‘235 294 4:17 4-250 000 
"117 647 2G 8:500 000 ‘250 000 1:4 4:000 000 
"125 000 IE: 8:000 000 ‘263 158 a) 3°800 000 
133 333 2:15 7500 000 ‘266 667 4:15 3°750 000 
142 857 17 7:000 000 272727 onvkb 3°666 667 

