Simultaneous equations involving the un lmowns to degrees higher than the first may sometimes be solved. Consider, for the pair of equations: ax + by + c se+ fxy gx hy k =0; from the former y=- (c ax) : a; substituting that y-value for y in the second given equation, a quadratic ins is found; this gives two substituting these in the given linear equation, the two corresponding values of y are found. The corresponding values must be properly paired: thus the equations 3x + 4y -5=0 and 2x'- sy y'- 22 =0 give x=3 and - 109:53, and 148:53; the proper pairing is x=3, and .r =- 109:53, y =148:53; the equations are not satisfied by x=3, y=148:53, for example. Once more, the two quadratics + 3xy=24 xy + give, on division (member by member) and clearing of fractions, 2 Cr'+ 3xy) =7 (xy + 4y') whence x = 4y or - 7y :2. For x =4y, the second given equation furnishes 4y' + and y=1 or -1, whence x=4 or -4; using x =- 7y :2 in like manner, one finds y= + 4 or -4 and x =- 14 or 14; in all four pairs of values corresponding thus: x= 4, y=1; x=-4, y=-1; x=14, x 14, y=4. In general, an equation of nth degree and one of nth degree in two unknowns are both satis fied by mn pairs of numbers. The solution of such a pair involves, in general, the solution of an equation of degree mn.
Permutations and Any arrangement (in a row) of r things (regarded as belonging to a set of n things) is called a (straight) permutation of the n things r at a time. Two permutations are the same when and only when they consist of the same things in the same order. The number of different (possible) permutations of n things r at a time is often denoted bysPr. To find this num ber, think of any one of thesPr-i permutations of n things r -1 at a time. There remain n- r + 1 things. Put one of these after the things of the given permutation. There so re sults a permutation of the is things r at a time. It readily follows that sPr=sPr-i I) , ,Ps-s = Pr-a (n-r uPs = ss PI • •-• I), ISA n. Multiply ing these equations member by member, it is found that *Pr = n (n - I) . . . (n - r+1). If r=n, sPs=n1, where n! (or (ft) means 1 X 2 X 3 X X n and is read factorial n. It can be readily proved that the number P of permuta tions of is things (a, b, c,...) n at a time, p of the things being a's, q of them b's, ..., is . If the order in a permutation of V• • • is things r at a time be disregarded, the result is a combination of is things r at a time. Two combinations are the same if they consist of the 'same elements. A common symbol for the number of combinations of n things r at a time is NC,. By permuting the r things of a com bination in every way, r 1 permutations arise. It follows that sCr.rt =s Pr, whence sCr=sPr:rt Since, on taking r things from n things, there remain is - r things, it is seen that tiCr=sai-r.
Arithmetical Progression.-An A.P. is a series of numbers such that the difference be tween any two adjacent terms is the same as that between any other two adjacent terms. The general A.P. is: a, a + d, a 2d,..., a d. Th, theory involves five elements: the common difference, d; the first term, a; the last, 1; the number of terms, ft ; and the sum of the terms, s. Given any three of the
elements, the remaining two can be found. Since ,C, =10, there are but 20 problems to solve, giving rise to as many formula:. The formula for 1 in terms of a, d and n obviously is l=a To find s in terms of a, 1 and n, let s= a+ (a + d)+ (a+2c1)+ + (I - 2d) + (l - d)+ l; then s =1 + d) + (I- 2d) + .. . + (a - 2d) + (a - d) + a; adding, 2s = is (a +1), whence s 2 (a+1). The remaining 18 for mulm, completely exhausting the subject, are: 2s - id + V 2ds+ (a - id)'; l = - a; = s (n 2 - s = (26 + id); s 1 + a 2 as • s = (21 - a =1 a = (n - 1)d; a = id ),/ (1 + - 2ds; 2 a -1; d = (1 - a) + (ts - 1); d = 2(s - an) n (n - 1); d (P - (2s -1 - a); d = 2 (111- s) -i- (n - 1); n= 1 + (1 - a) + d; n = (d - 2a -± V (2a - d)' + 8ds) 4- 2d; n = 2s + (a + 1); n (21 + d V (21 +d)0 - 8ds ) 2d.
Geometric Progression.-A G.P. is a series of numbers such that•the ratio of any one to the next is equal to the ratio of any other one to its next. Accordingly, the general form of a G.P. is: a, ar, ar',. . . , arm- I. Again, there are five elements to be considered: the first term, a; the last, 1; the ratio, r; the number of terms, n ; and the sum of the terms, s. In terms of any three of the five elements, either of the remaining elements can be expressed. Accordingly the theory of the G.P. involves the solution of but 20 problems. Most im portant of these are the problems, to express 1 in terms of a, r and n, and to express s in terms of a, r and n. It is plain that 1, or the nth term, is r. To find s, let s=a+ar+ ... thence rs = at+ ars + . . . + ars; substracting, and dividing by 1-x, it is found a(1 --rn) = a - that s The remaining 18 1-r 1-r formula: are easily obtained. If r be nu merically less than 1, the G.P. is said to be a decreasing G.P.; otherwise, not. In case of a decreasing G.P., it is possible to sum the series to infinity, a phrase requiring explanation. An endless series, . , is said to be infinite, i.e., to contain an infinite number of terms. A series that has an end, a last term, is finite. Let sn denote the sum of the first n terms of an infinite series. If the series such that there is a finite number L from which, by taking is large enough, ss may be made to differ by less than any prescribed amount and to which ix, as n continues to increase, approaches nearer and nearer in value, then L is named limit of ss as n increases endlessly, the series is said to be convergent (see SERIES) and L is called the sum (to infinity) of the series. Observe that here the word sum is used in a new sense, viz., as limit of a sum (in old sense). Now consider the infinite G.P., a, ar, are, arm, The sum s of the first n terms, by the forego ing formula for s, is: ss = a - . If the 1-r 1-r G.P. is a decreasing one, r