INDETERMINATE ANALYSIS. 



may be olitiined. But as thefe queftions generally admit 

 ct a great number of folutions, the objcft of enquiry is 

 not fo much to find the values of the i .termediate quan- 

 tities, as to determine a priori the number of them that the 

 equation admits of; and this, therefore, fliall form tlie fub- 

 jecl of our future invelligation. 



Now we have feen, that in the equation 

 ax + by — c, 

 the number of folutions is generally expreffed by 



b a' 

 f and q being firft determined by the equation 



ap — bq = I. 

 If, therefore, in the equation 



a .V + b y =■ J ~- c z, 

 ■ive make fuccefllvely z = I, 2, 3, 4, ?cc. the number of fo- 

 lutions for each value of z \vill be as below ; viz. 



, rr, ■ <^d-c)p {J-r)q 

 ax + by =d — c, number 01 lolutions — i 



ferved, that when the number of terras do not confift of a^ 

 exacl number of periods of circulation, the remaining 

 terms or fraftions mull be fummed by themfelves, which is 

 alfo readily effefted, as they will be the fame as the leading 

 terms of the firft period ; and it muft alfo be remembered, 



that - is to be confidered as a fraction in the firft feries ; 

 b 



but not - in the fecond, as is explained in the foregoing pro- 



pofition. 



Ex. I.— Given J v + 7 ^> + 11 z = 224, to find the 

 number of folutions that the equation admits of in pofitive 

 integers. 



Here the greatefi limit of 



20: alfo 



+ byz= J— 2i 

 + by-(l-^c 



( J-2c)p _ { d—2c' q 

 * ■ ■ • b a 



{d-i^_{d-3c)q 



- • ' • i a 



&c. S:c. &c. &c. 



the fum of which will be the total number that the gi»-en 



equation admits of ; and therefore, in order to find the 



exatl number of folution? in any equation of this kind, we 



ift firft afcertain the fum of all the integral parts of the 



the equation 5/ — 7 j = i, we have/> = j, and q = 2, 

 alfo a =; 5, and b = y ; and therefore the two feries, of 

 which the fums are required, beginning with the ieaft term 

 in each, are 



7 7 7 7 7 



] 2. 4 2. i; 2.26 2.37 ^ ^___ 2. II, 



c r-^^^' 



arithmetical feries 



{ld^^(!tLl±+i±:ii}P+i±:^^l)l+Scc-, and 

 J b b b b 



I (illfl? + ^±:.^f}!} 4. (i=L3£)? ^ 'J^a£)J ^ Sec 



the common difference in the firft. being - — —; and ia 



the fecond — ; alfo the number of terms in each 20. 



5 

 Whence we have 930 for the fum of the firft ; 

 and - - 868 for the fum of the fecond. 

 Again, the firft period of fraftions, in the firft feries, is 



b ' b 



and 

 be the cxaft number of 



427 

 7 7 7 



the difference of the two w 

 intregal folutions. 



Now in both thefe feries, w3 knew the firft and laft 

 term, and the number of terms ; for the general term being 



(d 



)P 



and 



f./- 



•z) q 



we ftiall have the extreme terms by taking the extreme li- 

 mits of z, that is z = I, and s < — ; which laft value of z 



alfo exprefles the number of terms in the feries. 



Hence then, having the elements of the progreflions 

 given, the fum of the wliole leries is readily obtained; 

 and if therefore we alfo find the fum of the fractional parts 

 in each, we fhall have, by deducting it from the whole 

 Turns, that of the integral part of the feries as required. 

 The latter part of this problem is readily effefled, for the 

 dcnominatoi' in each term being conflant, the fraftion.'; 

 .will neceftarily recur in periods ; and the number in each 

 can never exceed the denominator: it will therefore only 

 be ncceffary to find the fum of the fractions in one period, 

 which being multiplied by the number of periods, will give 

 the fum of the n-aftional part of the terms ; and thefe 

 taken frpm the total fum, will give the fmn of the in- 

 tegral part of the feries ; awd then, from what has been 

 b. fore obfcrved, the difference of the two fums will be the 

 jiuniber of integral folutious required. It may alfo be ob- 



and in the fecond feries, the firft period of fraftions is 



3 + o + i+i -f i=a; 

 5 5 5 5 



— being confidered as a fraftion in the firft, but not 



- in the fecond. 

 5 



Now the number of terms in each feries being 20, we 

 have 2 periods and 6 terms in the firft feries = 2x4 + 

 the firft 6 fractions =11 for the fum of all the fraflions ; 

 and therefore 930 — 1 1 =; 919, which is the exaA fum 

 of the integral terms. And in the fecond, we have 4 pe- 

 riods = 4.4 = 8, and therefore 868 — 8 = 860, which 

 is the fum of the integral terms in this ; awd hence accord- 

 ing to the rule 



919 — 860 = 59 ; 

 wliich is the number of integral folutions. 



Remark. — Simpfon, in his algebra, makes the number of 

 folutions to this quellion 60 ; but he has evidently intro- 

 duced one (w'z. .v = 10, y = 14, and z = 14,) which 

 does not obtain. 



Ex. 2. — Having given 7 a- + 9J1 -f 23 z = 9999, it is 

 required to find the number of its folutions in pofitive 

 integers. 



Here the greateft limit of z < ^ ■ = 434 ; alfo in 



the equation "j x — ^ p = ij we have / = 4) and q = $t 



a =■ , 



