VARIATIONS 87 ordinate axes respectively. Jellett remarks l that, if the proposed integral had been / , we should then have arrived at the equation = I cos a- -- + co.sS ), and consequently have seen that the two p /JL dx dyj curves contained under the equation p"=f(x,y,y) are such that, if one renders / /j.ds a maximum or a minimum, the other possesses the same property with regard to / 14. Next, to find the curve such that the surface generated by its revolution round a given line shall be a minimum. Here, neglecting a constant multiplier, we may write U= I y/l+y-dx ; J .Co accordingly by (15) we have = cVl + #-. This gives -- = ; Vy - c- c x + b ,-./ + vi < / " +c ^ / (* )> which represents the common catenary. 15. We now proceed to generalize the results in 11. If Fbe n function of x, y, y, . . . ?/ (n) , not containing y explicitly, we have hv integration P _ 2 _)_ 3 1 dx dx" dx n ~ .(20), where c is an arbitrary constant, Again, if F"does not contain x explicitly, ( V) = yP+ i/P l + . . . + ?/(" + 1 )/ ) )i . Also, from the equa- flp , " lion M= we see that P=r ax ... + (D? - ( - 1 where D l represents the differentiation of y, y, &c., and l, / .,, &c. Consequently, by integration that of (21), a differential equation of the order In - 2 at the highest. Again, if / contains neither x nor y explicitly, we may substitute c for (PJ, and consequently we have . +y M P,, ............ (22). 16. As an example, let us investigate the curve for which the area between the curve, its evolute, and the extreme radii of curva- /"i FHl + y 2 ) 2 ture shall be a minimum. Here U I pds= I -^-dx. Hence J ,s-,, J x y
iy (22), since yP^ = - J r , we have V Cj -f c.,/y, or p - r- 1 H 2 ;
ds Vl + z? 2 VI +y- thercfore -5 =c 1 cos <j> + c. 2 sin <f>, where (p is the angle made by the tangent with the axis of a-. This leads to s = - Cj sin <p + c. 2 cos (p + c 3 ..................... (23), which shows that the curve is a cycloid. Again, (1) if the extreme |ioints be fixed, the equations at the limits are (/ > i! ) = 0, (P 2 )i = 0- This shows that the radius of curvature vanishes at both limits and the curve must be a complete cycloid. (2) If the extremities lie on the curves y =f (x ), yi=fi(%i) , then the equations furnished Hence the extreme points are cusps on the cycloid; moreover, since in this case F = Q and Fi = 0, we have ?/ -fo(x Q } = 0, //] -/ 1 (o; 1 ) = 0, which show that the cycloid at each of its extremities touches these curves, and that the line joining the extreme points is normal to each of the bounding curves. It is easily seen that the minimum area is four times that of the circle which generates the cycloid. T 17. Next let us consider the variation of U I Vdx, J X where F is a given function of y, y, z, y, z,y,z,... y( n 2^ m iind y, z are undetermined functions of x. - dV n dV Let - * * 5 U= I * l Vdx + ( PSy + P 1 D8y + ...+ P n D"dy)dx then 1 roceeding as in 7, we find d U= I" (P)Sydx + r Q)dzdx + r,, J x a 1 Calculus of Variations, p. 140. (25), where (P), (P-^, . . . have the same meaning as in 7, and (<?), (Q^), . . . are the corresponding functions for the variable z. f* 1 18. The determination of y and z where I Vdx is a
- o
maximum or a minimum leads, as in 8, to the equations (/ J ) = 0, (Q) = 0, along with the equation at the limits P n Sy( n -V + (Q 1 )Ss + ...ft,,^" 1 - 1 )} =0. The mode of treatment is similar to that for a single de pendent variable. 19. In the discussion of a curve which possesses a maximum or minimum property, if we limit the investiga tion to all curves of a given length or which satisfy some other condition, we have a distinct class of problems, which originated in the isoperimetrical problems of James Ber noulli. They were originally styled isoperimetrical, but are now called problems of relative maxima and minima. Thus, let it be proposed to determine the form of y which renders fXl U = I Vdx a maximum or a minimum, and which also Xo /?! satisfies the condition U I Vdx = constant, where V and J V are given functions of x, y, y, &c. It is obvious that, if U is a maximum or a minimum, so also is U+alf, where a is any arbitrary constant. Accordingly the problem re duces to the determination of the maximum or minimum f* 1 value of the integral I ( F+aW)dx, regarding a as a con- J 2-0 stant, whose value is to be determined by aid of the given value of U . 20. Another class of problems closely allied to the pre ceding is that in which the variables are connected by one or more equations of condition : for instance, when y, y, y, <fec., z, z, &c., are connected by a relation TT=0, we investi gate the maximum or minimum values of U= r (V J *o where A is an indeterminate function of (26), We will illustrate these principles by examples. 21. To find the curve connecting two fixed points such that the surface generated by its revolution round a fixed line shall be given, and the volume of the generated solid shall be a maximum. Taking m n the fixed line as the axis of .r, we may write U= I (y J i + ay Hence by (15) we liave ?/ 2 + yVl +y~ = c + VI ?/ .(27). If the curve meets the fixed axis we have c = 0, and the curve is a circle whose centre lies on the axis. The further integration of (27) depends on elliptic functions ; it can, however, be shown without difficulty, as was proved by De- launey, that the curve is that generated by the focus of an ellipse or hyperbola which rolls on a fixed right line. 22". As an illustration of the method of 20 we shall consider again the problem of 13, taking the arc for independent variable. Let x = -,- , & = -, then we have the relation :>- 2 4- y- - 1 = ; and we write ds ds U= I u. + iX(.r 2 + y" - 1 )} ds = I Vds, where V= n + i(.r 2 j*o . y Hence for a maximum or minimum we have tli ere fore or dfj. _ . dfj. . dfj. _ d
ds dx dy ds