so that the value is the same as it would be if x were put instead of x; that is, the value is (x') if x' be less than 1, and (21-x) if x be greater than 7. It is obvious that for any negative value of x the value is the same as for the corresponding positive value. Similarly we may shew that if x is positive and = 2ml+x', the value of the right-hand side of equation (4) of Art. 314 is the same as if x were put instead of x, and is (x') if x be less than 1, and - (27-x) if x be greater than 1. And for negative values of x the value is the same numerically as for the corresponding positive value, but with an opposite sign. 312. It may be observed that in the fundamental demonstration of Art. 313, we suppose that when h approaches unity as a limit, the expression however large n may be. We may shew that no error arises from this supposition, by proving that the latter integral vanishes when n is increased indefinitely. For (v) cos Nπ (v - x) 18 (v) dv = sin πη which shews that the integral on the left-hand side will vanish when n is infinite, at least if o' (v) be not infinite. 313. We have not yet alluded to one of the most remarkable points in connexion with the formulæ (3) and (4) of Art. 314. In these formulæ (x) need not be a continuous function; for example, from x=0 to xa we might have (x) = f(x), then from x = a to x = b we might have (x)=f(x), then from x = b to xc we might have (x) = f(x), then from x = c to x = 7 we might have $ (x) = f(x). The formula (3) for instance would still be true for all values of a between 0 and 7 inclusive, as is evident from the mode of demonstration, except for the values where the discontinuity occurs. When for example xa, then the value of the right-hand member would not be ƒ (a) or ƒ, (a) but {f(a) + ƒ2 (a)}• If therefore for x = a we have f(x) = f(x), the formula holds also when x=a. 314. Find an expression which shall be equal to c when x lies between 0 and a, and equal to zero when x lies between a and l. Take formula (3) of Art. 314. Here v = a, and then from v = a to v = (v) = c from v = 0 to zero; thus 7 it is this will give c when x = a. Or we may use formula (4) of Art. 314. Then this gives 0 when x = 0, and c when x = a. 315. Find an expression which shall be equal to kx from x=0 to x= and equal to k (1-x) from x= 2 = 2 other case, and π2n2 2 {2 cos cos nπ- 1}. Nπ when n is of the form 4r +2, and 0 in every = = If we denote this by y then from x = 0 to x both inclusive y=kx, then from x = to xl both inclusive y=k(1-x); for values of a greater than 7 the values of y recur as shewn in Art. 320. Thus the value of y is the ordinate of the figure formed by measuring from the origin equal lengths along the axis of x to the right and left, and drawing on each base thus obtained the same isosceles triangle. 316. As another example we may propose the following: find a function (x) which shall be equal to x from x=0 to a, then be equal to a from x = a to x=π-α, and then be equal to T x from x=π-α to x = π. x = The result is this is true from x = 0 to x=π both inclusive. 317. The student may verify the following examples. If x be numerically less than a the expression is equal to a x if x be positive, and a +x if x be negative. This may be obtained from Art. 315 by integration; or from equation (3) of Art. 314. suppose 7 to increase without limit; then if (v) be such that we have 262 CHAPTER XIV. APPLICATION OF THE INTEGRAL CALCULUS TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 319. WE will here give a few simple examples of the application of the Integral Calculus to questions relating to mean value and to probability. Let p (x) denote any function of x, and suppose x successively equal to a, a+h, a +2h, ... a + (n-1) h. Then $(a) +$(a+h)+4(a +2h) +...+${a+(n−1)h} n may be said to be the mean or average of the n values which (x) receives corresponding to the n values of x. Let b- a = nh, then the above mean value may be written thus [$ (a) +$ (a + h) +$ (a + 2h) +...+${a + (n−1) h}] h Suppose a and b to remain fixed and n to increase indefinitely; then the limit of the above expression is |