198 
THE RESOLUTION OP 
series l + 3 + 6 + 8 + 
7 
V = 
Quot. 
N.Ans 
0 
0—2z 
0 
1 
1 
5—2 2 
oi 
3 
O 
10—2-5 
5 
6 
0 
15—25 
7f 
8 
4 
20—22 
10 
1 1 
5 
25—22 
12+ 
13 
&c. 
&C. 
&c. 
&c. 
sion 110112: to which ad 
have 110113, for the nur 
eluding those whereih the 
last must therefore be fou 
ll + 13, &c. continued to 
297 terms; which terms 
(setting aside the tirst (be 
ing united in pairs, we 
shall have the arithmetical 
progression 9 19 + 29 
&c. where the number of 
terms to be taken being 
1-18, and common diffe 
rence 10, the last term will 
therefore be 1479, and the 
sum of the whole progres 
ing (l) the term omitted, we 
her of all the answers, in- 
alue of x is negatjve; which 
d and deducted. 
In order to this we have already found, that these ne 
gative values do not begin to have place till q is greater 
than 190: let, therefore, 191,192,193,«xc. be substituted, 
successively, for <7; from 
whence it will appear that 
the number of all the said 
negative values is truly 
exhibited bv the arithme 
tical progression 4 1-11 + 
18 + 25, &c. continued 
to 29G—190 terms; where 
of the sum is 39379; which 
subtracted from 110113, found above, leaves 70734, 
for the number of answers required. 
7 
Quot. 
X.Ans. 
191 
32 1 1 
4 
192 
32 32 
H>t 
1 1 
193 
32—53 
171 
18 
194 
32—74 
24 + 
25 
&CC. 
&c. . 
&c. 
&c. 
After the manner of these two examples (which il-» 
lustrate the two different cases of the general solution, 
given in the preceding problem) the number of answers 
may be found in other equations, wherein there are 
three indeterminate quantities. But, in summing up 
the numbers arising from the different interpretations 
of q, due regard must be had to the fractions exhibited 
in the third column expressing tire limits of ^; because, 
to have a regular progression, the terms of the series in 
the fourth column, exhibiting the number of answers^