9. Let OA and OB be two perpendicular radii of a circle, and P a variable point of the circumference between A and B. Drop a perpendicular PM to OA, and write the six trigonometric functions of the angle POA. Show what limits they approach: first, as P approaches 4; second, as P approaches B. Ans. The results may be tabulated: 46. Limiting Cases of Roots of Quadratic Equations.-We have seen that the roots of a quadratic equation are ax2+ bx+c=0 -b±√b2-4ac We can write each of the roots in two equivalent forms by rationalizing the numerator: We will investigate the changes occurring in these roots when one (or more) of the coefficients decreases toward the limit zero. Evidently (I) if c approaches zero, one of the roots approaches 一只 a the other approaches zero; and (II) if b and c both ap proach zero, both roots approach zero. III. Let a approach zero. Then That is, one root approaches the finite limit, the other in creases indefinitely. IV. Let a and b both approach zero. Then That is, both roots increase indefinitely. Applications of III occur in the simultaneous solution of a linear equation and a quadratic, if it happens that on eliminating one of the variables we obtain an equation of the first degree. In such a case we understand that of the two roots ordinarily to be expected one has disappeared through becoming infinite; that is, a slight change in the coefficients of the original equation would cause this missing root to appear as a very large quantity. If the equation that results from the elimination is in the form of a constant equated to zero, IV shows that both of the roots ordinarily to be expected have become infinite. The principles III and IV can be deduced in another way. Dividing the original equation by x2, we have: Examples. The distances of a point P from two given perpendicular straight lines are x and y respectively. Find these distances from the relations given in each of the examples below: 1. 3x2-xy+ y2+6x+6y=0\ x=0,y=0 y=2x 18 2. 3x2-xy+y2+6x+6y=0\x=0, y=0 y=-x 3. 2x2-3xy-2y2+3=0} Jor x=-13, y= — 36. twice. x=2y+1 Jor x=∞, y= ∞. 4. 2x2-3xy-2y2+3=0x=∞, y = ∞ x=2y twice. 47. Permutations and Combinations.-The different orders in which a number of things can be arranged are called their permutations. For instance, the permutations of the three letters a, b, c are: abc, acb, bac, bca, cab, cba. The different arrangements of the same letters taken two at a time are: ab, ba, ac, ca, bc, cb. Each distinct group which may be selected from a number of things, regardless of their arrangement, is called a combination. Thus, of the six permutations of the three things a, b, c taken all together, there is but one selection or combination of three different things, abc. The number of combinations of the same things taken two at a time is only three, ab, ac, and bc. 48. Number of Permutations of n Different Things. This is the same as the number of ways in which n things may be arranged in a row of n places. We may fill the first place in the row with any one of the n things; that is, in n different ways. In each case we may fill up the second place in (n-1) ways, i. e., with any one of the (n-1) things left, making n(n−1) different ways in which the two places may be filled, since we may follow any one of the n ways by any one of the (n-1) ways. Similarly we may fill the third place in any one of (n-2) ways, and the three places in n(n-1) (n-2) ways. Finally, we can fill the nth place in only one way, as we shall have used the (n-1) things in filling the (n-1) preceding places. Hence, there are n(n-1) (n-2)....3.2.1=n different permutations of n things taken all together. This is formulated nPn=\n, n." the symbol n being called "factorial n. 49. Permutations of n Different Things Taken s at a Time.— Counting the number of permutations of n things taken s at a time is the same as finding in how many different ways we may fill up a row of s places with n things; the number is represented by We may fill up the first place in n ways, the first and second in n(n-1) ways, the first three places in n(n-1) (n-2) ways, just as in the preceding article, until, when the last place, number s, is filled, we shall have or n(n-1) (n-2)....(n−s+1), P ̧=n(n−1)(n-2)....(n−s+1). If we multiply and divide this expression by the remaining (n-s) factors, it reduces to as the formula for finding the number of permutations of n things taken s at a time. Examples. 1. In how many ways can a crew of 8 men be seated in an eight-oared shell? Ans. 40320. 2. How many six-letter arrangements can be made of the letters causing? Ans. 5040. 3. How many four-letter arrangements can be made from the letters causing? Ans. 840. 4. How many four-figure numbers can be made out of the digits 1, 2, 3, 4, 5 without repetitions? How many five-figure numbers? How many numbers? Ans. 120, 120, 325. 5. How many of the integers between 50,000 and 60,000 can be formed from the digits 1, 3, 5, 7, 9 used without repetitions? Ans. 24. 6. How many permutations of 9 letters each can be made from 9 different letters if 3 of the letters must always come together in a given order? Ans. 5040. 7. How many permutations of 6 letters each can be made from 9 different letters, of which 3, if they are taken at all, must occur together and in a fixed order? COMBINATIONS. Ans. 1200. 50. Number of Combinations of n Different Things Taken s at a Time.-Suppose there are a combinations each containings things; from each combination there can be made [s permutations, and from the x combinations as permutations, which 51. If we find from this formula the number of combinations of n things taken (n-s) at a time, we obtain |