I N D 



INDERVA, a fmall ifland in the Perfian gulf; co 

 /....usW. of Ormus. 



INDETERMINATE, in Geometry, is underftood of a 

 quantity cither of time or place, which has no certain or 

 dL-finite bounds. See Indefinite. 



Indeterminate Analyfu, is a particular branch of 

 algebra, in which there is always propofed a greater num- 

 ber of unknown quantities than there are equations; whence, 

 from what we have feen under the article Alcebra, the 

 quellions become unlimited ; but in this fpecies of equations, 

 tlie folutions mull always be given in integers, or rational 

 fraftions ; and this condition frequently fixes a limit to the 

 number of anfwers that an equation admits of, and even 

 fometimes renders the problem impoffible ; though, in the 

 generality of cafes, the number of folutions is indeii- 

 nite. When, amongll the unknown quantities, there are 

 none that exceed the fimple power, the equation is faid to 

 be of ihejirjl degree : when the fecond power enters into the 

 equation, it is faid to be of \.hs fecond d'gree : and when the 

 third power or cube enters, it is of the third degree ; and 

 fo on. 



Every indeterminate equation of the firft degree, as 

 ax±by±cz±d±icc. = o 

 is folved by means of the more fimple equation 



ap — bq = -V I. 

 And every indeterm.inate equation of the fecond degree has 

 its folution, if not abfolutcly depending, at lead intimately 

 connecled with the folution of the equation 



^'-N?'=± I. 

 Before, therefore, entering upon the general folution of in- 

 determinate problems, it wall be proper to confider more 

 particularly the two equations above-mentioned. 



Prop. I. 



To find the values of x and y, in the equation 

 ax — by — iz 1. 



Firft, it may be obferved, that this equation is al^^-ays 

 poflible under either condition of the ambiguous iign +, 

 provided that a and b be prime to each other ; and without 

 this, the equation is always impoffible ; becaufe, in that 

 cafe, the firft fide of the equation will be divifible by the 

 common divifor of a and b ; whereas the fecond fide, + i. 

 Las no divifor ; and, confequently, no equahty can obtain. 

 But when a and b are prime to each other, the folution may 

 always be obtained by the following rule : Divide the 

 greattft of the two given numbers a and b by the other, 

 and then always the laft divifor by the laft remainder, as in 

 the ufual method of finding the greateft common meafure of 

 two numbers, and let the fucceffive quotients arifing from 

 this operation be denoted by ^, /S, 7, 0, &c. See Common 

 Meafure. 



With thefe quotients, placed in one horizontal row, make 

 the following calculation ; and t he terms of the laft frac- 

 tion but one will be the values of x and y required ; thus 

 quot'a ^ y S &c. 



i' (3 ' y/3-l-i ' (y/S-i- i)i + /3 

 which may be othcrwife exprefled in words : thus, having 

 arranged tlie fucccfuvc quotients, as above, the firft fradlion 

 will have a for its numerator, and 1 for its deiioir.inator ; 

 the fecond will have k p -|- i for its numerator, and ,S for its 

 denoniinator ; and all the other numerators will be found by 

 multiplying the numerator of the laft fraSion by the fol- 

 lowing quotient, and adding thereto the preceding numera- 

 itti, and the denominators are found exactly in the fame 



I N D 



manner, as is evident in tlie foregoing fractions. See 

 Ratio. 



Ex. I.— Find the values of x and j', in the eqisation 



l6.r — 41 J' = 1. 

 Firft 16)41(2 

 '32 



1)16(1 

 9 



7)9(' 

 7 



~)7(3 

 6 



,)2(2 



fra£lions -j — , 



quotients 2, I, I, 3, 2 

 i 3 5^ i^^ 41 

 ' 7 ' 16 



Here the laft fraAion but one is — , and therefore this gives 



the values of x and y ; that is, .i- = 18 and y = 7, which 

 renders 16 ;t — 41 j^ =; l, as required. 



■Ex. 2. — Find the values of .v and _)•, in the equalioa 



17*- ij^= - ,. 

 Firft ,5)17(1 



'5 



~)i5(7 



~)2(2 

 2 



Quotients I, 7, 2 •" 



Fraftions J -, -, — 

 1 1 7 >5 



Therefore x = 7, and_y = 8, which gives 

 17. r - isy = - I- 



The demonftration of this rule belongs properly to con. 

 tinned fraftions, which the reader will find very ably treated 

 of, in the Englifh edition of Euler's Algebra ; and in moil 

 of the French writers on that fubjedt, particularly in tiie 

 »' Eflai fur la Theorie des Nombres, par Legendre." 



Having thus fhewn the general method of folving the in- 

 determinate equation a.v — iy =z -r 1 ; it only remains to 

 make a few obfervations relating to it, and to fliew, that one 

 folu.ion being obtained, an infinite number of other folutions 

 may be deduced from the one known cafe. 



In the above examples we found a .V — iy = -f- i, and 

 a .X — by z= — I, as th<- queftions required ; but we are 

 freq '.!•..':'.' ly led to the folution a .t — by — + i, when the 

 queftion requires — i, and thy contrary; which fcems at 

 fivft to 'leftroy the generahty of the rule; but this difficulty 

 is eafily furmounted from the following confidcrations. 



Let aj> — b J =: + I, the values of/iand q being known, 

 to find, from this equation, the value of .x and y, in the 

 equation nx -~ by = — 1 . 



Since ap — bq= I, wc have only to make x-=lm~f, 



and 



