INDETERMINATE

In Mathematics, quantities which change their values are said to be indeterminate, in opposition to those whose values remain fixed and invariable. For example, in the equation of the ellipse bix- + a2yi = (fl 42, tho co-ordinates x and y are indeterminate, while the axes a and b are constant, quantities. Indetkrminate. In Botany, when a stem is never terminated by a flower, nor has its growth stopped by any other organic cause: example, Veronica arvensis. INDETERMINATE ANAEYSIS, is a branch of Algebra which has for its object the investigation of problems which admit of an infinite number of different solutions. When the enunciation of a problem furnishes a le number of equations than there are unknown quantities, the problem is indeterminate in this sense— that the equations may be satisfied by a igning to the unknown quantities an infinity of different values. It happens, however, in the greater number of cases, that the nature of the question requires the values of the unknown quantities to be expre ed in whole and positive numbers; a condition which greatly restricts the number of solutions. As in the ordinary algebraic analysis, problems belonging to the indeterminate analysis are of the first or second degree, according as the simple powers or squares of the unknown quantities enter into the equations expre ing their conditions. Every equation of the first degree, containing two unknown quantities, may be reduced to the form ax-fby — c; and it is plain that if only one equation of this sort is given, the values of a; and y are wholly indeterminate, for any value whatever may be given to one of these quantities, and it will be easy to find a value of the second which will satisfy the equation. In the indeterminate analysis, the object is to find all the values which x and y can have in whole numbers; the constants a, b, and c denoting whole numbers, either positive or negative. For example, let it be proposed to divide the number 159 jsto two parts, one of which is divisible by 8, and the other by 13. If we denote by x. and y the quotients of the required parts, divided respectively by 8 and 13, these parts themselves must be 8 a; and 13 y; consequently, the problem Is expre ed by this equation, 8x + 13j/ = 159 (1), aud it will be solved when x and y are expre ed by whole pos i'ive numbers., 159 — 13?/ In the first place, this equation gives a: =, or 7 5,1 r =19 —!/-] -— -; but as x and y must be whole numbers,it is obviously nece ary that — o- — be also a whole 7 — 5« number. Suppose this to be n; we have then — - — = »; whence 5y-r-8m = 7.... (2), an equation of the same form with (1). The viilue of y deduced a a irfrom this,iisj/: = 7-——- —8n, or t/ = 1 — n-\ 2 — —3n • Now, precisely in the same2 3jjmanner as before, since y and 7i are whole numbers, — a-— must nece arily be a whole number. Suppose this to be?;', and we have = »'; whence 3n-f-5n' = 2.... (3.) „From this...last equaUon.we get «= 2 — 5 re', orre = — re + 2— 2 re' Continuing the same proce , we now make • 3 2 — 2re' = re" (where n" must be a whole(4); number), whence!re' + 3re" = 2, -"in" from which we obtain re' = -, or re' = 1 — n" — Supiosc, next, — =n"', and there results 7i" = 2re'".... (5). Thus at last.in equation has been found which is satisfied by making n equal to any whole number whatever; and, on attending to the different steps of the proce , it will readily be observed that x and y, and the indeterminate quantities re, re', re", re'", are connected by the following system of equations: 598 [s. 612]