INDETERMINATE ANALYSIS. 



« = 7, and i = 9 ; alfo 9999 — 23 . 434. = 17 ; tkcrrfore Prop. V. 



the feries whofe fums are required, will be u- • u r .■ 1 r .u .t. 



^ Having given any number or equations, lets than the 



., 4 ■ '7 4-40 4 • ^i . (. 4 ■ 997^ number of unknown quantities that enter therein, to detcr- 



9 99 i-c.... ^ ^j^^ thofe quantities^ 



3.17 3 • 40 3.6' ^ . 0976 ^°' ''"-"'■'^ '^^ propofed the two equation* 



*J- -7- + -7- + -^ + &c. . . . ---— t a X + t y + c ^ = d 



•&\t common difierence in the firil being 1-1-il = 10 — ; '« fin<l tl^= values of .r, ji, and z. 



9 9 Multiply the firil by a\ and the fecond by a, whence, 



a.d in the fecond ^-^ = 9 ^ : alfo the number of terms ^^ f^btradion, we obtain 



7 7 (a'i — a i*) j; + (y <; — a <:) 2 = a' </ — a rf' 



in each 434 ; that being the grcateft limit of 2. Or, dividing each of thefe known co-efficients by their greateft 



Hence we have the fum of the firft feries = 963769^ common divifor, if they have any, and reprefenting the refukt 



9 by b\ c", and d", this equation becomes 



and of tlie fecond ... - 929349 h'' jr + ^" z = d' . 



Alfo the firft period of fraftions in the firft feries, is Find now the values of y and % in this equation ; and thefe 



being fubllituted for them in the equation 



579246813 ^ , ^, 



999999999 x= ^ 



»nd— =48periods,and 2terms= J. 48 +~+ -=241 - will give the correfponding values of jf, of which thofe, 



9 9 9 9 of courfe, mult be rejected that render x fraftional, and 



And in the fecond feries, the firft: period of fraftions is alfo thofe that give (c a + by) > d. 



£Ii 4.^4.!' Ill— J -^•'' Given the equations 



1 1 ^^ 1 1 ^ n 1 ~^' (-3 *■ 4- 5/ +7 = = 560 



jgg_ X9.r + 25J> + 49 a = 2920 

 to find all the integral values of *•, y, and s. 



? » Multiplying the firtl by 3, we have 



Hence 963769 1 - M> - =963528 integral terms .v + ijj, + 21 z = ,6Sc, 



and 929349 — i36 =929163 integral term* t9 ■*■ + ^5 J' + 49 = = 29-» 



. whence 



, ,.rr ^ • . . f • f 10 r 4- 28 a = 1240, or 



whence the difference 3436) is the number of m- -j / , ^ _ g^^ 



tegral folutions required. .,, ,,r 1 r . ■ 



In the foregoing examples we hare had two of the terms And here the values of jf and * are fourd to be 



prime to each other; but when this is not the cafe the fol- _y = no, 96, 82, 68, 54, 40, 26, 12 



lowing transformation will be neccflary. a = 5, 10, IJ, 20, 25, 33, 35, 4* 



Ex. 3. — Let there be propofed the equation, »nd of thefe, the only two that give 



12 jr -I- 15 J- f 20 « = looopi, ^ 560 — 7 s — 5j r 



to find the number of folutions. * ~ 3 



Here no two of the co-efficients are prime to each other, j^ integer, are as follows, 



and we muft therefore proceed as follows. _ — Rj — fc 



Divide the whole equation by 3, and tranfpofe e, and { z = ^o, v = 40', x = co 



^ _ J which are the only two folutions the equations admit ofin 



4* + J > = 33334 - 7 '^ -i- — r— integers. 



The method above given will never fail of producing all 



which laft muft be an integer; make therefore 1=-' =. u, ^\^ poffible folutions in equations of the above form; but 



" 3 there are other methods that may be followed m particular 



and we have z = 3 u + 1 ; fubftituting now this value of cafes which fometimes (horten the operations. Thefe the 



z, the original equation becomes reader will find explained in vol. ii. of Eulcr's Element* of 



12 X 4- i;.r + 20 (3 " + = loocoi ; Algebra, 



or, dividing by 3, Prop. VI. 



4 ■*' + 5 J" + ^° " = 333*7' To decompofe a given numeral fraftion, having a conb- 



the number of folutions in which will be the fame as in the pofite denominator, into a number of fimple fradiens having 



equation propofed, which will be foimd as in the foregoing prime denominators. 



examples, except that here the leaft value of « = o, becaufe -p^is is, in fad, only an application of the foregoing 



we fhall then Itill have z = i, and by proceeding as in 00 



the preceding examples, it will be found that the number prapofitions to this particular problem ; for let — bo tht 



of integral (olutions that may be given to. this qucftion, " 



amounts to no lefs than 138861 1. given fraftion ; and fuppofe, in the firft. inftanoe, that it*. 



Vol.. XIX. G ■ d«no. 



