INDETERMINATE problem, is that which admits of many different solutions and an. swers, called also an unlimited problem. In questions of this kind, the number of unknown quantities concerned is greater than the number of the conditions and equations by which they are to be found; from which it happens, that generally some other conditions or quantities are assumed, to supply the defect, which, be ing taken at pleasure, give the same num ber of answers as varieties in those as sumptions. If, for instance, it were re quired to find the value of two square numbers, whose difference is equal to a, a given quantity. Here if x. and y= de note the squares, then which is only one equation for finding two quan tities. Now, by assuming some other un known quantity, as z, so that z=x+y= z the sum of the roots; then is x=. -± a for x.—y2=a x'+2xy+y'=z.
z = V.-Fa z'±a 2x+ 2z. • And by the same mode Y=511-2-; which are the two roots having the difference of their squares equal to a given quantity a, and are expressed by means of an as sumed quantity z; so that there will be as many answers to the question, as there can be taken values of the indeterminate quantity z.
Mr. Leslie, in the transactions of the Royal Society of Edinburgh, has given a paper on this subject, the object of which is to resolve the complicated expressions which we obtain in the solution of inde terminate problems into simple equa tions, and this is done by means of a principle, which, though extremely sim ple, admits of a very extensive applica tion. Let A x B be any compound quan
tity equal to another, C xD, and let m be any rational number assumed at pleasure; it is manifest that, taking equimultiples, A xm B=C xm D. If, therefore, we sup pose that A = m D, it must follow that m B C, or B Thus two equations of a lower dimension are obtained. If these be capable of further decomposi tion, we may assume the multiples n and p, and form four equations still more sim ple. By the repeated application of this principle, an higher equation, admitting of divisors, will be resolved into those of the first order, the number of which will be one greater than that of the multiples as sumed. For example, resuming the pro blem at first given, viz. to find two ration al numbers, the difference of the squares of which shall be a given number. Let the given number be the product of a and b; then by hypothesis, a.z—y.=ab; but these compound quantities admit of an easy resolution, for x-Fy X x—y aX6. If therefore we suppose x Ina, we shall obtain m, where m is arbitrary, and if rational, x and y must al so be rational. Hence the resolution of these two equations gives the values of x and y, the numbers sought, in terms of tn, m'a±b Ida. x , and