Tilt HfcSOLUTlON OF 
136 
q<7 — 2z, and 5 \- 7q — respectively. From the 
former of which the greater limit of q is given 
or 15 7 and from w ^, expressing the 
lesser limit, we have 61, for the value of <7, when the 
least value of a’ becomes equal to that of y. These 
limits being assigned, let q be now interpreted by 0, 
I, 2, 3, 4, 5, &c. successively, up to 61, inclusive; 
whence the number of answers, or variations of y cor 
responding to every interpretation, will be found as in 
the margin. From whence it appears that the arith 
metical progression 4 + 11 -f 18 f 25 f 32, con 
tinued to 62 terms, will truly ex 
press the number of all the an 
swers when q is less than 62: 
which number is therefore given 
= 4 f 61 /7 -f 4 X 31 = 13485. 
In all which answers it is evi 
dent, that x, as well as y, will 
be positive (as it ought to be): 
because, it has been proved that 
the least value of x, till q be 
comes ( — —^) — 6ly, Will be greater than that 
c 
oft/; which is positive, so far. But now, to find the 
answers when q is upwards of 61, we must have re 
course to the general value of x; which, in these cases, 
by the different interpretations of z, becomes negative 
before that of y. Here, by beginning with the greatest 
limit, and writing 137, 156, 
155, 154, &cc. successively, 
in the room of q, it will ap 
pear, that the number of 
answers will be truly ex 
pressed by the series 4 + 8 
4- 13 -f 17 4- 22, ¿cc. con 
tinued to 157 —61 terms: 
which terms being united in 
pairs (because, in every two 
terms, the same fraction in the limit of z occurs) the 
series 12 -p 30 -f 48 -f See. thence arising, will be a 
9 
X __3 
Z — 
N. Ans. 
157 
9 — 2 5 
41 
4 
156 
18 — 2 3 
9 
8 
155 
27 —Qz 
13* 
13 
154 
30 — 2z 
18 
17 
153 
45 — 2Z 
22t 
22 
&C. 
Sec. 
&C. 
&C. 
7 
y - 
N. A ns 
0 
5 — 2 
4 
1 
12 — z 
H 
2 
19 —2 
18 
3 
26 ^ 
25 
4 
33 — 2 
32 
Sec. 
&c. 
&C.