Full text: A treatise of algebra

500 RESOLUTION or IN jpLTERM IN ATE PROBLEMS. 
By proceeding, therefore, as in the aforegoing cases, 
we have x — 8331 — y— * ; whence the least va- 
lue of y is given n 3, and the greatest of x — 6328. 
Moreover, from the equation 4 m f 5n - 20, we have 
yi zz b— n —whence n zz 0, and m — 5. 
Therefore the general values of x and y [gken in Problem 
13) do here become 8328 — 5q — 5u, and 3 4q; 
from the former of which the greatest limit of q is given 
8328 
= -—-= 1665. Now, since the value of y will here 
continue positive, in all substitutions for q and u (as 
no negative quantity enters therein); the whole num 
ber of answers will be determined by the values of x 
alone. 
In order to this, let q be successively expounded by 
1665, 1664, 1663, &c, 
and it will thence appear 
that the said number will 
be truly defined by 1666 
terms of the arithmetical 
progression \ f 2 f 3 
-f 4 +- 5, &c. whereof 
the sum is found to be 
J38S611» 
7 
X 
Quot. 
N.Ans. 
1665 3—bU 
1664 8—5 U 
1663 13—5U 
&C. j &C. 
Of 
If 
2| 
&c. 
1 
2 
3 
&C. 
When there are four indeterminate quantities in the 
given equation, the number of all the answers may be 
determined by the same methods: for, any one of those 
quantities may be interpreted by all the integers, suc 
cessively, up to its greatest limit (which is easily de 
termined); and the number of answers, corresponding 
to each of these interpretations may be found, as above; 
the aggregate of all which will consequently be the 
whole number of ans\yers required: which sum, or 
aggregate may, in many cases, be derived by the me 
thods given in Section 14,* for summing of series’s by 
means of a known relation of their terms. But. this 
being a matter of more speculation than real use, 1 shall 
now pass on to other subjects.
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.