24 INFINITESIMAL CALCULUS whence we get x = V3a 2 + b 2 - 2b ; neglecting the negative root, which is inadmissible. If b>a, +J3a 2 + b*-2b is negative, and accordingly this gives no solution in the case of an exterior planet. For an interior planet we have a>b ; and it remains to deter mine whether /3a 2 + b 2 - 2b lies between the maximum and mini mum values of a;, i.e., between a + b and a &. Since a > b, it is immediately seen that Jd* + b 2 - 2b is > a - I- The remaining condition requires a + b > V3a 2 + b- -2b, or a + 3b i. c., a 2 + Gab + Qb 2 > 3a 2 + b 2 , or Hence we easily find b > -7- . We accordingly infer that this gives no real solution for a planet nearer to the sun than one-fourth of the earth s distance. When this condition is fulfilled it is readily shown that the corresponding solution is a maximum, and that the solutions corresponding to x = a + b and x = a - b are both minimum solutions. 66. Many problems of maxima and minima contain two variables, which are connected by an equation of condition. Thus, to find the maximum or minimum values of <t>(x, y), where x and y are connected by the relation /to y)-o. Here we have d$ d$ rfy =() ty^df % = 0> dx dy dx dx dy dx Accordingly the maximum and minimum solutions are obtained from the simultaneous equations
- .f^ = 0, and/fa, y)-0.
dx dy dy dx More generally, if n + 1 variables, x,x l) x z ... x n , be connected by n equations F^O, F 2 = 0, . . . F,, = 0, and it be proposed to find the maximum or minimum value of a given function f(x, x, x 2 . . . a;,,) of these variables, we have, by differentiation, the equations dx n dV n dx dx i which give, on elimination, the determinant equation df df_ _ _ _ df_ dx dx { dx n dx dx l dx n dx dx l dx n This joined with the given equations determines the system of values of x, a^ , x 2 . . . x n for. which the function may have a maxi mum or a minimum value. Maxima and Minima for Functions of two or more Independent Variables. 67. Let u = <j>(x, y), then, as in 64, if x and y are inde pendent, the maximum or minimum value of u must satisfy the equations dit, A , du , -5-=0, and-j-=0. dx dy Suppose x and y Q to be values of x and y which satisfy these equations; then, in order that they should correspond to a real maximum or minimum value of u, the expression <(>(x + h, y + k)- <p(x , y ) must have the same sign for all small values of h and k, as in the former case. Again let A, B, C be the values of . . ^respectively, dx 2 dxdy dy 2 when x*=x and y = y . Then, by 57, <;>( + h, y n + k)- <f>(x , y (} ) = ( A7i 2 + 2B7j& + C& 2 ) + &c. But, when h and k are very small, the remainder of the expansion is, in general, very small in comparison with A7t 2 + 2B7iA;+C/L 2 ; and consequently the sign of ^>(a? + h, y + k)- <f>(x , ?/ ) depends on that of Now, in order that the latter should have the same sign for all small values of h and k, AC - B 2 must not be negative ; i.e., must not be negative. When this condition holds, the resulting value of u is a maximum .when A is negative, and a minimum when A is positive. The necessity for this con dition was first established by Lagrange. In the particular case where A = 0, B = 0, C = 0, then for a real maximum or minimum it is necessary that all the terms of the third degree in h and k in the expansion of (f>(x + h, y + k) should also vanish, and that the quantity of the fourth degree should pre serve the same sign for all values of h and k. The preceding discussion admits of a simple geometrical inter pretation by considering the surface represented by the equation z = <f>(x, y) ; since it reduces to finding the points on the surface of maximum or minimum distance from the plane of xy. 68. Next let u=(f>(x, y, z), where x, y, z are independent variables. Here, as before, if x , y , z correspond to a maximum or a mini mum value of u they must satisfy the equations ~~$ dx Accordingly we have , _ ~^ s=: U. "^ dy dz = i (Ah 2 + BP + CP + 2Ykl + 2GAZ + 2HAA) + &c. , where A, B, C, F, G, H represent the values of d 2 ii d 2 u d z u d 2 u d z u d^u dx* dy* dz? dydz dzdx dxdy respectively, when x = x , y = y n , z = z . Now, as in the former case, in order that u should have a maxi mum or a minimum value, it is necessary that A/<, 2 + B 2 + CP + 2Fkl + ZGlh + 2HM- should preserve the same sign for all small values of h, k, and 7. If we multiply by A, this expression may be written (Ah + H& + GZ) 2 + (AB - H 2 )& 2 + 2(AF - GH)W + ( AC - G 2 )Z 2 . Consequently the sum of the last three terms must be always positive. Hence, in order that the expression in question should be positive for all small values of h, k, and I, we must have A>0, A, 11, >0. a, h h, b A, H, G II, B, F G, F, C This result is also due to Lagrange. The corresponding conditions for the case of four or more inde pendent variables can be likewise determined, and are readily ex pressed in the form of a series of determinants. See Quarterly Journal of Mathematics, 1872, p. 48. (1) To find the maximum or minimum value of the function ax 2 + 2hxy + by* + 2gx + 2fy + c. It is easily seen that when h^>ab, there is neither a maximum nor a minimum value. When ab >h 2 , and a >c, we obtain, as the minimum value, a, h, (j h, b,f ff,f, c {2} Similarly it can be shown that the maxima or minima values of ax* + by* + 2hxy + 2gx + 2fy + c a x^ + b y 2 + 2h xy + 2g x - are the roots of the cubic equation a - a u h - h u h - h u b - b u g-g u f-f u (3) Of all triangular pyramids standing on a given triangular base, and of given altitude, find that whose surface is the least. Tangents and Normals to Curves. 69. The infinitesimal calculus furnishes, as we have seen, a gene ral method of find- ing the tangent at any point in a curve whose equation is given. For example, let y=f(x) be the equation, in Cartesian coordinates, to any curve ; and suppose (x, y), (x l ,y l ) g g u f-f u c-c u =0. _. r to be the coordinates of two points F,Q in the curve (fig. 5), and
