so REESE LIBRARY OF THE UNIVERSITY OF CALIFORNIA /tT//>__ Shelf No. RESEARCHES IN THE CALCULUS OF VARIATIONS. PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. RESEARCHES CALCULUS OF VARIATIONS, PRINCIPALLY ON THE THEORY OF DISCONTINUOUS SOLUTIONS: , TO WHICH THE ADAMS PEIZE WAS AWAEDED IN THE UNIVEESITY OF CAMBEIDGE IN 1871. I. TODHUNTEE, M.A. F.B.S. \\ LATE FELLOW AND PRINCIPAL MATHEMATICAL LECTURER OF ST JOHN'S COLLEGE, CAMBRIDGE. r u// ' ILon&on anU Cambridge: -'ft A*/ \ MACMILLAN AND CO. 1871. [All Rights reserved.} OA3I: T57 PREFACE. THE subject of this Essay was prescribed in the following terms by the Examiners : THE ADAMS PRIZE. A determination of the circumstances under which disconti- nuity of any kind presents itself in the solution of a problem of maximum or minimum in the Calculus of Variations, and applica- tions to particular instances. It is expected that the discussion of the instances should be exemplified as far as possible geometrically, and that attention be especially directed to cases of real or supposed failure of the Calculus. E. ATKINSON, Vice- Chancellor. J. CHALLIS. A. CAYLEY. J. CLERK MAXWELL. CLARE COLLEGE LODGE, April 21, 1869. It was after much hesitation that I resolved to discuss the subject; the fact that it had given rise to some controversy, how- ever, naturally led me to enforce what I believed to be the correct and adequate explanation of the difficulties which had been raised. The Essay then is mainly devoted to the consideration of discontinuous solutions : but incidentally various other questions vi PREFACE. in the Calculus of Variations are examined and I think elucidated. I entertain no doubt of the substantial accuracy of the theory here developed ; but at the same time I am aware that in an extensive investigation, which is original and somewhat abstruse, there may be a few subordinate statements which require to be restricted or corrected. I indulge the hope, however, that on the whole I shall have definitely contributed to the extension and the improvement of our knowledge of this refined department of analysis. The Essay is published as it was originally written ; with the exception of the mistakes in algebraical work which occur occasionally in manuscript, but are rendered evident in the clearer form of print. Also a few references have been supplied, and some short remarks, chiefly on passages to which attention had been drawn by the Examiners : these slight additions are enclosed within square brackets. The laborious task of correcting the press has been undertaken for me by my friend the Rev. J. Sephton; and as on other occasions I am much indebted to him for thus aiding me with his knowledge and accuracy. I. TODHUNTER. CAMBRIDGE, 26th October, 1871. CONTENTS. CHAPTER I. PAGE MAXIMUM OR MINIMUM OF AN INTEGRAL WHICH INVOLVES ONLY ONE DIFFERENTIAL COEFFICIENT 1 CHAPTER II. THEORETICAL INVESTIGATIONS 13 CHAPTER III. DISCONTINUITY PRODUCED BY CONDITIONS . . . .32 CHAPTER IV. MINIMUM SURFACE OF REVOLUTION . , . . ... . 54 CHAPTER V. MAXIMUM SOLID OF REVOLUTION 68 CHAPTER VI. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V 100 CHAPTER VII. BRACHISTOCHRONE UNDER .THE ACTION OF GRAVITY . . .126 CHAPTER VIII. PROBLEM OF LEAST ACTION . . . . . .147 viii CONTENTS. CHAPTER IX. PAGE SOLIDS OF MINIMUM RESISTANCE. , , . .167 CHAPTER X. SOLID OF MINIMUM RESISTANCE WITH GIVEN VOLUME . . 196 CHAPTER XL JAMES BERNOULLI'S PROBLEM . . . . , v . 220 * CHAPTER XII. MULTIPLE SOLUTIONS . . . , . . . . . 243 CHAPTER XIII. AREA BETWEEN A CURVE AND ITS EVOLUTE . . . . 250 CHAPTER XIV. MISCELLANEOUS OBSERVATIONS . 259 UNIVERSITY OF CALIFORNIA. CHAPTER I. MAXIMUM OR MINIMUM OF AN INTEGRAL WHICH INVOLVES ONLY ONE DIFFERENTIAL COEFFICIENT. 1. IN order to arrive at a knowledge of the circumstances under which discontinuity occurs in problems of the Calculus of Variations we shall discuss various problems, beginning with some which are extremely simple. We shall find that speaking gene- rally discontinuity is introduced by virtue of some restriction which we impose, either explicitly or implicitly, in the statement of the problems which we propose to solve. We do not define the word discontinuity; but we shall always make it obvious in what sense we use it as we proceed. In our investigations we shall never ascribe any variation to the independent variable but only to the dependent variable : in this respect we follow the practice of some of the most eminent authorities on the subject. [Although a preference is thus expressed for the method of treating the Calculus of Variations which has been adopted by Strauch and Jellett, yet it must not be supposed that this is of importance for the following pages. The results are not affected by this circumstance, although the investigations are rendered in some cases more simple and intelligible than they would otherwise have been.] 1 2 MAXIMUM OR MINIMUM OF AN INTEGRAL 2. Let p stand for -=* ; and let $ (p) denote a given func- tion of p : required the curve for which the integral l< (p) dx taken between fixed limits is a maximum or a minimum. Let u \(p)dx, then, as far as terms of the second order inclusive, - *' (P) &y Then we require by the usual theory ' therefore $ (p) = constant .................. (1). The term outside the integral sign in the value of 8u vanishes, since the extreme points of the curve are supposed to be fixed. Thus SM reduces to | [<}>" From (1) we obtain constant values of p ; for any such value " (p) if it does not vanish will be permanently positive or perma- nently negative throughout the limits of the integration ; thus in the former case we have a minimum value of the integral, and in the latter case a maximum value. 3. We may remark here that it is very important to avoid the common error of using the words ike greatest value when we ought to restrict ourselves to the words a maximum value, and the words the least value when we ought to restrict ourselves to the words a minimum value. In the present essay we shall use the words greatest and least, and other similar terms, only when they are strictly applicable. 4. Now return to equation (1). The required curve must be rectilinear ; as the extreme points are given the value of p is WHICH INVOLVES ONLY ONE DIFFERENTIAL COEFFICIENT. 3 known : thus the value of the constant in (1) is determined. Therefore we have apparently only one solution, which furnishes a maximum or a minimum according as $" (p) is negative or positive. But a little consideration shews that it is quite possible to have cases of the problem which require another solution. For instance "(p) may be negative for the known value of p, and yet it may be obvious that there must be some line straight or curved for which the integral has its least value ; in fact such a statement must in general be true for any form of the function <. 5. We may naturally ask then if the constant in (1) must necessarily have the same value throughout the limits of integra- tion. Suppose, if possible, that ' ( p) is equal to C v for one part of the required line, and equal to (7 2 for the remaining part. Then the integrated term '(/>) = <> (2). Every condition of the problem may then be satisfied ; at least in many cases. Thus suppose we take two values j^ and p z found from (2), and draw the corresponding straight lines, one through one of the given points, and the other through the other given point. Then if "(pj) and <"(#j) are both negative we obtain a maximum; and iffifj and "(p^ are both positive we obtain a minimum. But if " (PJ and " (p a ) are of opposite signs we do not obtain either a maximum or a minimum. For $u reduces to \ *" (?,) /(%>)& + \ " (A) /()'<**. 12 MAXIMUM OR MINIMUM OF AN INTEGRAL where each integral extends between the limits of x which belong to the corresponding value of p. We may suppose p to vanish if we please through one of the two portions into which our integral is divided : thus in this way we can make &u have which sign we please ; and therefore with these values of p there is neither a maximum nor a minimum. [Instead of taking ' (p) = the more general form <' (p) = C ought to have been taken ; for this may lead to discontinuous solu- tions by furnishing different values of p. The discontinuity would be like that illustrated in Art. 9.] Let us now take some particular cases. 7. I. Then Suppose < (p) p (1 +p z j. Here '(p) cannot vanish. The only solution is the straight line which joins the two given points ; and this makes the integral a minimum : for we may suppose that p is positive. Moreover as there must be a least value of the integral in this case, it is certain that the minimum value which we have obtained is the least value. No maximum presents itself in this case. In fact we can make I (p) dx as large as we please. Thus in the diagram let A y c and B be the given points ; take A for the origin, and from B draw BG perpendicular to the axis of x. Then < (p) vanishes along AC, and is infinite along BG\ and we can draw a curve WHICH INVOLVES ONLY ONE DIFFERENTIAL COEFFICIENT. 5 very near to AC and CB for which / (p) dx will be as great as we please. But we do not obtain a maximum in the technical sense of that word. We can indeed get a greater result by making our curve go below A G. Difficulties might be suggested as to this case. For as x does not vary along CB it might be said that I (p) dx along CB must vanish, since the limits of the integration coincide. If however we transform I p (1 +p 2 ) dx into I dy (1 +p*) we obtain an integral in which the limits do not coincide. 8. II. Suppose (f> (p) = ^ a , and that the straight line which joins the fixed points makes with the axis of x an angle less than 60. Here the straight line which joins the two given points cor- responds to a maximum, for we may suppose that p is positive. When (j) (p) = we have p = 1 ; and these two values give opposite signs to "(p), so that we do not obtain either a maxi- mum or a minimum by combining these two straight lines. But p = oo is also a solution of ' (p) = ; and it will be found that by combining p = 1 with p oo we obtain a maximum, and in fact the greatest value of the proposed integral. No minimum presents itself in this case. In fact we can make I $ (p) dx as small as we please. For taking the diagram of the preceding Article we have (p) vanishing along AC and CB : and we can draw a curve very near to AC and CB for which l(p}d& will be as small as we please. But we do not obtain a minimum in the technical sense of the word ; the integral can be made to change sign as.it passes through zero. MAXIMUM OR MINIMUM OF AN INTEGRAL 9. III. Suppose = Z> 2 - 3/>* - a 2 . Then <' (p) =p(p*~ a 2 ), <" ( Here the straight line which joins the two given points cor- responds to a maximum or a minimum according as the value of p z is less or greater than -^. The solution made up of the two straight lines AC and BG which are determined by p = a corresponds to a minimum. Or we may form a solution by combining more than two straight lines, if for every one of them p 2 = a 2 , so that p a. The value of the integral is the same whatever be the number of tacks comprised in the solution. In continuing the discussion we will for simplicity put a= 1. Then in this example if the angle between BA and Ax is greater than 30, we get a minimum either by the straight line from A to B or by a tack. We may remark in passing that it is rare to obtain two minima solutions of a problem in the Gal- WHICH INVOLVES ONLY ONE DIFFERENTIAL COEFFICIENT. 7 culus of Variations, or rather has been hitherto rare : we shall see other examples. To determine which of these two corre- sponds to the less result we must determine whether 6 2 ^+ A ~ 2 4 is greater or less than 6 3 - + -j , that is whether p* 2p* 4- 1 is greater or less than 0; it is obvious that (p 2 I) 2 is greater than 0: hence the value of the integral is less for the solution with the tack than for the solution which consists of one straight line. We may put (p) in this form 2 1 1 and then it is obvious that the least value of I (p) dx is always obtained by supposing p* = 1. If the angle between BA and Ax is less than 30 there is only one minimum solution, namely that with the tack. No maximum presents itself except in the case in which the angle between BA and Ax is less than 30. 10. In the preceding Article the straight lines which form the tacks are equally inclined to the axis of x : but this need not necessarily be the case in other examples. See Art. 8. If the equation ' (p) = furnishes us with roots numerically unequal we may have straight lines forming tacks which are not equally inclined to the axis of x. This plurality of solutions is well illustrated hereafter in the solid of minimum resistance with a given surface. If $ (p) is always positive and finite and cannot vanish, (p) dx must be susceptible of a minimum value. Suppose that <' (m) = and that " (m) is negative, then p m does not give us a minimum. In this case the equation ' ( p) = must have two other roots besides m, one greater and one less than m. This is easily illustrated geometrically, supposing p the abscissa and (j> (p) the corresponding ordinate of a curve. 8 MAXIMUM OR MINIMUM OF AN INTEGRAL The point M is that which corresponds to ' (m) = and " (m) negative. It may happen, as in the lower diagram, that p= for the other roots of <' (p) = 0. cases 11. Thus we see that the discontinuity which occurs in some of the problem of making /< (p) dx a maximum or a mini- mum is that of two or more straight lines meeting at an angle. 12. A particular case of this problem has been considered in a paper on the Brachistochronous Course' of a Ship: see Philoso- phical Magazine for January 1834; pages 33... 36. This paper is I believe the first in which the interesting kind of discontinuity we are here considering was noticed ; other kinds of discontinuity had already been discussed, as for instance some by Legendre. We may suppose the axis of x to coincide with the direction of the wind. The velocity of a ship may be supposed to be a func- tion of the tangent of the angle which the direction of the ship's course makes with the direction of the wind ; and from the nature of the case this function must be an even function, so that we may denote it by /Q? 2 ). Thus we require the minimum value of the WHICH INVOLVES ONLY ONE DIFFERENTIAL COEFFICIENT. 9 integral \ *, 2 f dx, the limits being supposed fixpd. ' // ^ / ,, Put (p) for v ^"^ ; ; then we get , ( 4y /, - f ( where / /', and /" are used for brevity instead of f(p*), f'(p*), and /"(/). 13. We may observe that /(^> 2 ) will in general involve radi- cals with ambiguities of signs ; for the velocity will not be the same for two angles between the ship's course and the direction of the wind which are supplemental, although p* will have the same value for the two angles. Suppose, for example, that the velocity varies as the square of the cosine of half the angle between the ship's course and the direction of the wind ; then f(p*) varies as ' where the upper or the lower sign must be taken according as the angle between the ship's course and the direc- tion towards which the wind blows is less or greater than a right angle. We may observe also that ^(I+p*) must be taken negatively in the integral I /./ 2 ^ dx whenever x is algebraically de- creasing. In order that the value p = may correspond to a minimum we must have/(0) 2/'(0) positive. 14. I shall now make some remarks on the problem which has hitherto been discussed ; the second and the third remarks I consider of peculiar importance, for as we proceed it will be found that they are applicable to many other problems. I. It may be said that in a certain sense there is no discon- tinuity ; for example in Art. 9 we obtained two straight lines meeting at an angle ; now two straight lines might be regarded as a, conic section, that is as forming but one curve. I lay no stress 10 MAXIMUM OR MINIMUM OF AN INTEGRAL on this remark, merely introducing it to shew that it has not been overlooked. In fact the solution resembles the double solution of a quadratic, from which indeed it arises. II. Such discontinuity as occurs may be said to be introduced by the conditions which we impose on the problem. We require a line to have a certain minimum property and to connect two fixed points; and the discontinuity arises from the circumstance of there being two fixed points. Suppose we take the following pro- blem : find a line such that Iff) (p) dx may be a maximum or a minimum, the line commencing at a given point and having a given length. Here by the usual theory we have to find a maxi- mum or minimum value of I j (p) +\ V(l + p 2 )f dx, where \ is some constant to be determined. Thus we obtain = constant; and in order that the term in the variation which is outside the integral sign may vanish this constant must be zero. Also 4>(p) +WO- +^ 2 ) must be zero at the limit of the integration which is not fixed. Thus we can eliminate X and we have for determining p the equation Then the unfixed limit of integration will be determined from the fact that the length of the line is given. Here no discon- tinuity presents itself. The last equation may furnish us with more than one value of p, but we shall not be able to combine two different values into one solution, unless indeed the two values of p which we employ give the same value to -/ ^ , and also give Vl +p* . , (q) denote a given function of q : required the curve for which the integral I (q) dx taken between fixed limits is a maximum or a minimum. Let u \ dx which is required to be a maximum or a minimum, where < is a known function of y and its differential coefficients with respect to x. Change y into y -f 8y ; then in the usual way we obtain for the variation of the integral to the first order an expression of the form I M By dx, where L depends on the values of the variables and the differ- ential coefficients at the limits of the integration. Now if 8y may have either sign we must have M=. as an indispensable condition for a maximum or a minimum ; and moreover we must also have L 0. These statements are universally admitted to be true. Suppose however that owing to some condition in the problem we cannot always give to Sy either sign: for example suppose that throughout the whole range of the integration Sy is essentially positive, then it is no longer necessary that M should vanish. If M is positive through the whole range of the integration we are sure of a minimum ; and if M is negative through the whole range of the integration we are sure . of a maximum. We assume of course that we are able to satisfy the condition L ; or to ensure that L shall be positive in the, former case and negative in the latter case. 14 THEORETICAL INVESTIGATIONS. Next suppose that y may have either sign through part of the range of the integration, but that it is essentially positive during the remainder of the range. Then if M vanishes through the former part and is positive through the latter part of the range we are sure of a minimum ; and if M vanishes through the former part and is negative through the latter part of the range we are sure of a maximum. We assume as before that the condition relating to L can be satisfied. Now we must observe a great peculiarity in the case which we are considering; when Sy does not vanish throughout the range for which its sign is restricted we are sure that the varia- tion of the integral is essentially positive or essentially negative without examining the terms of the second order in the varia- tion. 18. Simple as the remark is which is the subject of the pre- ceding Article it will furnish the foundation for much that will follow: in fact it is the principle on which depends the discon- tinuous solution of nearly all the problems we shall have to discuss. The principle appears to have been first employed by Mr Todhunter in the Philosophical Magazine for June 1866. For the applications we shall have to make of the principle we may state it thus : Suppose we are seeking by the aid of the Calculus of Variations the curve which has some assigned maximum or minimum property; then if no condition is imposed which fetters the sign of y there can be no solution except such as may be supplied by putting J/= 0. But suppose a certain boundary is imposed which the curve is not to pass beyond ; then along that boundary By will not be susceptible of both signs, so that part or the whole of this boundary may occur in the required solution. Thus the solution can consist of nothing besides what can be ob- tained from M 0, or of part or the whole of the given boundary, or of some combination of these two elements. Many illustrations of this statement will occur as we proceed. 19. The integrals with which we shall be concerned will in general have to be taken between assigned limits. Hence the variation which we have denoted by L + I M By dx is more ex- THEORETICAL INVESTIGATIONS. 15 plicitly presented thus, where L l denotes the value of a certain expression at the upper limit of integration, and L Q the value of the same expression at the lower limit. Now suppose we separate our range of integration into two parts, one extending from x to f and the other from f to x^ Then the variation corresponding to the range from X Q to f may be denoted by and the variation corresponding to the range from f to x l may be denoted by If there be no discontinuity in the function to which variation has been given L z and L z will really denote the same thing ; but if there is discontinuity L z and L z will not necessarily denote the|same thing: and we shall have to be very careful on this point when we are considering the value or the sign of the whole variation. 20. A simple example may be here conveniently solved, as it will illustrate some of the remarks made in the present Chapter. Required a curve which shall connect two fixed points A and B on the axis of x, and make I ().= . where the subscript 2 relates to the point D as being on the arc AD, and the subscript 3 relates to the point D as being on the arc BD. The only way to make this vanish always [unless both q 2 and q 3 are zero] is to have q^q z , and also 3p 2 = Sp 3 : the latter requires that^ 2 =p a , for then, and then only, 8p 2 and Sp s mean the same thing. [This however implies that we assume there is to be no break of direction at D.] Hence we must have I* + 6aC+ 2(7' = | 2 + Gay + 2 7 ' ! - a +3a*C+ 2aC'+ G" =4 +3a 2 7 + 2a 7 / + 7 // . D O These equations with those which have been previously ob- tained suffice to determine all the arbitrary constants. THEORETICAL INVESTIGATIONS. 19 We have still left in the variation of the integral the term of the first order here Sy 2 and &^ 3 mean the same thing, so that the term reduces to 6(<7- 7 )Sy 2 . Now by the nature of the problem % 2 can never be negative. Hence we are certain of a minimum if C 7 is positive, for then this term is positive or zero ; and the term of the second order is positive. From (4) . 4 h 2 Similarly 7' = 87^ -y- . Substitute in the first of equations (6) ; thus C(a-h)= so that 07 = a + ft we have then to shew that C is positive. From (5) Substitute for C' ; thus h 3 b s-iii / . 7\ / r> /"Y7 l(> \ r-i t 2 1 L2\ T^WJ^T^V C (a+/2') 36/1+ -r } + C(a +ah+h)-\ ^-r- ,, \ 4/ !^TJ a ii that is 0"+C(a 2 -2aA- Similarly 22 20 THEORETICAL INVESTIGATIONS. , From (7) and (8) by subtraction that,. -7 But from the second of equations (6) we get G" - 7" = 2a (7' - 0') + 3a 2 (7 - C) ~a + h~ so that ^ a . T - C+ 2 2 12 a 2 #" 5A 2 -a 8 therefore 4 C (A a) = -7^ 2 TO fl Q> \.u Thus G is positive provided 25 is greater than (5 2 - a 2 ) (A 2 - a 8 ) 12 that is provided b is greater than a 4 a^ 2 5A* 24" 4 ' f 24 ; and this condition is satisfied inasmuch as D is supposed to be outside the curve (2). It will be observed that the solution is discontinuous; the branches which meet at D are determined by different equations ; at the common point p and q have respectively the same value for each branch. The result obtained is a minimum ; it is not the least value of the proposed integral. It is obvious that the integral can be made arithmetically as small as we please by taking parts of the axis of x and two THEORETICAL INVESTIGATIONS. 21 straight lines inclined to the axis of x at angles very nearly right angles : this is illustrated by the parts AG, GD t DH t HB of the diagram. [The result obtained is a minimum subject to the condition introduced in the investigation. We may enunciate the result thus : if any other curve be drawn by giving to y and q infini- tesimal changes, so as not to exclude D, and to have no break of direction at D, the integral has an algebraically greater value than it has for the assigned solution. We may briefly notice another case which presented itself. Suppose we put ^ = 0, and q z = 0. Then we obtain ~ a + h a h "Ta- 7= T2~ ; therefore ( G 7) Sy 2 = ~ Sy 2 . Hence this case does not give a minimum.] 21. It will be useful for the sake of reference to present some known formulae with respect to terms of the second order in a variation ; I shall add something to the usual investigations of them. I confine myself, as sufficient for my purpose, to the case in which the function under the integral sign involves no differ- ential coefficient higher than the first. Let then u = I (f> (x, y y p) dx, where as usual p stands for -j- . Change y into y + By ; then to terms of the second order inclusive we have 22 THEORETICAL INVESTIGATIONS. Denote the term of the first order in Sw by v, and the term of 1 2 the second order by ~ w ; then we see that w = and we may conclude that this relation will still hold after v has been transformed in any convenient manner. Now if we transform v in the usual manner, using the sub- scripts 1 and to denote values at the upper and the lower limits of integration, we obtain /d(f> j, \ fd w = - hr-$y tSrftfl + \\j -~T- (TT \dp y / t \dp V t J V,*/ flfo W We conclude then that w? is the variation of this expression ; that is we conclude that of which the last line may be written > d d* This is the form in which it is found convenient to put the term of the second order in the variation according to the known method of Jacobi. It is easy to obtain this form directly instead of indirectly as we have done ; as we will now shew. 22. We have By integration by parts we have THEORETICAL INVESTIGATIONS. 23 Also by integration by parts we have , r, a* , r, s N2 d ( d*\ , (Sy) Sy S 7 , cfo? - (oy r -7- , , ao? ; ^ J) ) u ^ dydp J ^ ' dx \dydp) Thus we obtain the required transformation of w. 23. We shall now suppose that the limiting of values of y are fixed, so that Sy l = and 8y = 0. Hence we have w = i r> i $*$ ^ f d* \ where P stands for -~ ,- 7 7 dy* dx\dydp) and Q stands for -~ . Let z be such a quantity that and assume 8?/ = tz ; then therefore 24 THEORETICAL INVESTIGATIONS. Since the limiting values of y are fixed the part of the last expression which is outside the integral sign vanishes. Hence finally the term of the second order in Su, Here z' is used for ~ ; and in like manner y will be used for ax This is Jacobi's form for the result. We see at once that there will be neither a maximum nor a minimum unless Q preserves the same sign throughout the range of the integration. 24. Suppose then that Q does retain the same sign through- out the range of the integration. We shall now consider what further conditions are necessary in order to ensure a maximum or a minimum : this is a point which the ordinary treatises on the subject seem to me to discuss in an unsatisfactory manner. I. Suppose that z can be taken so as never to vanish through- out the range of the integration ; then there is a maximum if Q is negative and a minimum if Q is positive. For we see that the expression under the integral sign in the term of the second order in Su is necessarily of the same sign as Q ; and it will not vanish unless throughout the range of integration z$p z'Sy = Q, which we may write thus zBy z'Sy 0. But this leads to 8y = Cz, where C is a constant, and as % vanishes at the limits z must also vanish, which is contrary to the supposition. Now as we shall see presently z is of the form C^yj + C 9 f 9 where (7 t and <7 2 are arbitrary constants, and/ x and/ 2 are definite functions of x. If f t and f 9 do not vanish nor become infinite within the range of integration we can secure that z shall not vanish. For *- c t c/i-nn/J. C -f where m = ? ; and will not range from positive infinity to i / negative infinity, but only between certain finite limits ; so that by ascribing to m any value outside these limits we secure that z shall not vanish. THEORETICAL INVESTIGATIONS. 25 II. Suppose that we cannot secure that z shall not vanish throughout the range of integration. Our assumption that by tz is not admissible if z vanishes when Sy does not vanish. Hence it might appear at first sight that the proposed method of trans- formation simply becomes inapplicable, and so leads to no result. But as we shall shew we can infer that there is now neither a maximum nor a minimum. For from what has previously been said it follows that ~ will now range through every value from /2 positive infinity to negative infinity. Take m such that z shall vanish at the lower limit of integration. Then by reason of the range of values of which ~ is susceptible z will also vanish at or /2 before the upper limit of integration. Take Sy = Cz, where G is a constant, for all values of the variable x between those for which z vanishes : and take Sy = Q for other values of x. Then the term of the second order in w vanishes. The term of the third order will in general not vanish, but will be susceptible of either sign. Thus there is neither a maximum nor a minimum. 25. Suppose that the value of y found from ................ ........... is denoted by f(x, c lt C 2 ) where c t and C 2 are arbitrary constants: then this value of y is of course the solution of the problem of the Calculus of Variations which is supposed to be under discussion. Equation (2) will also be satisfied when we give small arbitrary increments 5c t and 5c 2 to the constants. And as we shewed in Art. 21 that w = Bv we infer that (1) will be satisfied when we put And as (1) is linear with respect to z we see finally that the general value of z is ri df df C ^ C *^> where C t and <7 2 are arbitrary constants. 26 THEORETICAL INVESTIGATIONS. This finishes the exposition of Jacobi's method for discrimi- nating between a maximum and a minimum, so far as will be necessary for our purpose. 26. I propose however to consider more particularly the term of the second order in the variation of ly(f>(p)d(c. This is of course less general than the problem which has just been given after Jacobi ; but it includes a large number of particular cases, and it will furnish some results which have not hitherto been specially noticed. Let u then to terms of the second order inclusive we have >' (p)} dx = I Transform the term of the first order in the ordinary way, and suppose the limiting values of y to be fixed ; then this term becomes To make this vanish we must have and this leads to y*(p)-yjP*'Cp) = Ci ....................... (3), where c, t is a constant. Now consider the term of the second order. By integrating by parts we have Sy S P '(p)dx = (Sy)' f ( p) - JBy ^ [<' ( p) Sy] dx ; therefore 2 jSy Sp f (p) dx = (SyW (p) - /(%)' ^'(p)dx; THEORETICAL INVESTIGATIONS. 27 and thus the term of the second order is that is 1 Jf (p) [y (Spf - y" (SyY] dx. This supposes that By vanishes at the limits ; if it does not the term of the second order is where the subscripts 1 and refer to the upper and lower limits of integration respectively. At present we will however continue to suppose that By vanishes at the limits. 27. In geometrical applications we shall generally be able to regard y as positive. If then the curve given by (3) is concave to the axis of x we know that y" is negative ; and thus the sign of the term of the second order will be invariable if that of <" (p) is so. But we can shew that the sign of " (p) is invariable if that of y" is ; for from (3) we have therefore by differentiation c i P y so that thus we see that if one of the two y" and $' (p) is of invariable sign so also is the other. Thus if y be positive and the curve be concave to the axis of x we have a maximum if "(p) is negative and a minimum if "(p) is positive : and, in this case we need not have recourse to Jacobi's method. 28 THEORETICAL INVESTIGATIONS. 28. If however the curve be convex to the axis of x we cannot settle the sign of the term of the second order without further examination : to this we now proceed. From (3) we have where ^ (p) denotes a known function of p. Now x Thus * = CiX(p) + c t ........................... (5), where % (/?) is some definite function of p, and c 2 is an arbitrary constant. The value of y in terms of x is theoretically to be found by eliminating p between (4) and (5). We denote the result of this elimination by Although we cannot actually effect the elimination generally yet we shall be able to obtain the forms of ~ and ~f- which &1 dc^ are required in Jacobi's method. For from (4) and (5) we obtain The second of these equations gives S othat c = Again, from (4) and (5) we have THEORETICAL INVESTIGATIONS. 29 The second of these equations gives _ i - . dy so that -j 2 - = p. dc, Hence the quantity which we denoted by z in Arts. 23 and 25 becomes in the present case p(x-c,) that is c_ ) (Cj C, Q where m is a constant standing for -^ . If the expression just given between brackets vanishes at any point we have at that point a Now x - is the abscissa of the point of intersection of the P tangent to the curve at the point (x, y) and the axis of x ; we will put f for this abscissa. We assume that y is positive and that the curve is convex to the axis of x. 29. I. Suppose that the tangents at the extreme points of the curve, that is the fixed points, intersect above the axis of x. Then f is not susceptible of all possible values ; for instance, if the extreme points of the curve are on opposite sides of the lowest point f is not susceptible of values lying between the values it has for the extreme points. Thus we can take c a mCj so that it shall not be equal to any admissible value of f : so that we can secure that z shall not vanish throughout the range of integration. Hence by Art. 24 we are assured of the existence of a maxi- mum or of a minimum if "(p) retains an invariable sign. 30 THEORETICAL INVESTIGATIONS. II. Suppose that the tangents at the extreme points of the curve do not intersect above the axis of x\ then there will be neither a maximum nor a minimum. For if the tangents inter- sect on the axis of x we can make z vanish at the two limits of the integration. If the tangents intersect below the axis of x we can make z vanish at one of the limits of integration, and also at some other point within the range of integration. Hence by Art. 24 there is neither a maximum nor a minimum. Particular cases of this general result have been noticed before ; but not the general result itself. See Dienger's Grundriss der Variationsrechnung, 1867, pages 21 and 25. 30. It is important to observe that the transformation given by Jacobi for the term of the second order in a variation holds even if we do not suppose the term of the first order to vanish. For instance : let then taking the limiting values of the variables to be fixed we obtain x ...... . ..... (1). Now if we put M 0, we have for every system of values of $y (2). But even if we do not put M = the above general value (1) of Su still holds ; and in this case it may be possible that some particular value of By makes I MSydx 0, and then Su reduces as before to the form given in (2). For example take the case of a brachistochrone under the action of gravity. Here x being measured vertically downwards THEORETICAL INVESTIGATIONS. 31 Hence y has to be found from P dx This leads to Hence x* -f c a vers" - fivers- 1 -- Hence with this value of 2 the expression (1) holds for u whatever be the relation between a; and y. Suppose for illus- tration that we take a curve consisting of an arc of a circle fol- lowed by an arc of a cycloid which has its cusps in the axis of y, the two arcs touching at the common point. Then Bu takes the form given in (1). The part of iMSydx which corresponds to the cycloid vanishes ; the other part of I M&ydx does not vanish always, but it may vanish for some particular value of By. The term of the second order in Su retains the same form throughout. Of course it is possible to give special transformations of the term of the second order in a variation in special cases. Thus the transformation in Todhunter's Integral Calculus, third edition, Art. 377, applies to the problem there discussed, that is the brachistochrone. CHAPTER III. DISCONTINUITY PRODUCED BY CONDITIONS. 31. WE shall now discuss some examples in which dis- continuity is produced by conditions explicitly imposed on the problems. We begin with a very simple case. Imagine a rectangular court containing four rectangular grass plots. Required the shortest course from a given point A on one side of the court to a given point B on the other side, with the condition that the path is not to cross the grass plots. Of course A and B might be so situated that a straight line could be drawn from A to B without crossing a grass plot. But if A and B are not so situated the path will be discontinuous, consisting of two or more straight lines not in the same direction. DISCONTINUITY PRODUCED BY CONDITIONS. 33 For instance, in the diagram the shortest path may consist of three parts, namely AC, OE, and EB. In this example the Calculus of Variations will assure us that the path must be made up of straight lines ; and then we must determine by Geometry or the Differential Calculus what assemblage of straight lines will constitute the shortest path. 32. "We shall now modify the general problem of Art. 2, so as to introduce discontinuity ; and as an easy mode of enunciating the problem we will suppose as in Art. 12 that we are dealing with the brachistochronous course of a ship. We will now make the very natural condition that the ship's course is not to cross certain prescribed spaces ; these spaces we may conceive to be forbidden on account of rocks, or shoals, or hostile batteries. Thus, for instance, suppose that a ship is to pass from A to B in the shortest time without crossing the straight line 00 pro- duced indefinitely to the right, or the straight line OD produced indefinitely downwards ; these straight lines being parallel to the axes, and Ax being the direction of the wind. We know from the discussion in Art. 12 that the course be- tween any two points if no obstacle occurs is a straight line, or is 3 34 DISCONTINUITY PRODUCED BY CONDITIONS. composed of more than one straight line ; and thus we see that in the present case the course must be composed of various straight lines, including possibly portions of the boundary of the forbidden space. We must then examine the various suppositions that can be made. For example, the swiftest course may perhaps consist of the straight line from A to some point P in OD, the straight line from P to 0, the straight line from to some point Q in C, and the straight line QB. We should have of course to investigate the positions of P and Q, if such there be, which would make this course swifter than any other. 33. A general solution of the problem of the preceding Article for any form of the function denoted by < (p) in Art. 12 would be impracticable. But if a particular form be assigned to , it might be practicable to complete the discussion : of course the final result might depend on the situation of the point B and of the straight lines OC and OD. 34. We will however discuss a particular case with some detail. Let the velocity be the function of p which is denoted by r- -2 , so that $ (p) in Art. 12 stands for (1 +p*)*. Suppose Ax the direction of the wind ; let CD be an arc of a circle : and let the swiftest course be required from A to B with DISCONTINUITY PRODUCED BY CONDITIONS. 35 the condition that the course does not cross the circle, \vhich is supposed large enough to forbid the direct course from A to B. The restriction here imposed might present itself in practice owing to the presence of a hostile battery of a certain range. We have to find the minimum value of < (p) dx where < (p) = (1 +p*)%. Here Thus '(p) = has no solution except p = 0; [and (p) = (7 will furnish only one value of p\ : and so the swiftest course be- tween any two points is the straight line which joins them ; see Arts. 2 and 4 Hence it will follow that if P and Q be adjacent points on the arc the direct course from A to Q is swifter than that made up of the straight line AP and the arc PQ. We shall by this consideration arrive ultimately at the follow- ing result: Let APbe & tangent to the circle from A, and BR a tangent to the circle from B ; then the swiftest course consists of the straight line AP, the arc PR, and the straight line RB. y 35. The solution here given furnishes a very good illustration of the important principle of Art. 18. Put u I < ( p) 32 36 DISCONTINUITY PRODUCED BY CONDITIONS. then to the first order Now along AP and RB we have y" = 0, so that the part of $u under the integral sign reduces to only so much as arises from the elements hetween P and R. Now for such elements y" is negative, and By is essentially positive, so that this part of Bu is positive. Since AP touches the circle the value of <' (p) is the same at P whether we consider the straight line AP or the circle : thus the term <' (p) by will enter twice with equal value and opposite signs, and so vanishes. Similarly the term ' (p) By at R vanishes. Hence Bu reduces to a small quantity of the first order which is essentially positive ; and thus a minimum is secured. I here suppose that in comparing the proposed path with an adjacent path we vary the whole path: if however we do not vary PR but only the pieces AP and RB the value of Bu will still be positive, but it will be a small quantity of the second order instead of the first, namely, jT$" (p) (Bp)* dx. 36. We may observe that with the law of velocity adopted in Art. 34 the swiftest course in Art. 32, if the direct course is forbidden, will consist of the straight lines AO and OB. In this case, proceeding as in Art. 35, we find that Bu reduces to where p Q is the tangent of OAx, and^ the tangent of BOO; and By is the variation of y at the point 0. Now By is essentially positive, and p is by supposition greater than p l : thus Bu is a positive quantity of the first order, 37. It is obvious that the process of Arts. 34 and 35 does not require that the curve which bounds the forbidden area should necessarily be a circular arc ; any curve which is concave to the DISCONTINUITY PRODUCED BY CONDITIONS. 37 axis of x may be taken instead of the circular arc. Nor need the law of velocity be necessarily that expressed by (p) may be any function of p such that <' (p) = [or <' (p} = C] has no possible root or only one possible root. 38. As another example let the law of velocity be such that 22 4 (ft (p) = 5 2 _ -1- + P- : see Art. 9. This in fact requires that the '-t 4 expression for the velocity should be 8 4 Suppose A and B so situated that the swiftest course from A y to B when there is no obstacle consists of the straight lines A and CB which are determined by p = a. And suppose that a certain circular area is not to be crossed so that this course cannot be adopted. Required the swiftest course. It will be found that the swiftest course consists of the follow- ing parts : the straight line AP drawn from A to touch the circle ; then the arc PQ where Q is such that the tangent at Q is parallel to AG\ then the straight line QD which is part of this tangent; and finally the straight line D CB. That we thus obtain a minimum can be shewn as in Art. 35. If we pass from the assigned path to an adjacent path the varia- 38 DISCONTINUITY PRODUCED BY CONDITIONS. tion will be found to be a positive quantity : it will be of the first order if we vary the whole path, and of the second order if we vary all except the part PQ. 39. In the problems of Arts. 34 and 38 it may be granted that we have obtained minima results, but yet it may be asked, how do we know that we have obtained the least results ? I answer that the Calculus of Variations is immediately concerned only with maxima and minima values ; and it would have been sufficient to use the term minimum in these problems. Nevertheless we can see that there must be a least value in each problem, and that the path must consist of a straight line or lines with perhaps part of the boundary of the forbidden area. Then, on trying various combinations of these possible components, we may soon convince ourselves that there can be no least values except those which have been assigned. The kind of discontinuity which these problems furnish is that of straight lines and a curve which touch where they meet. 40. Suppose we seek for a curve of maximum or minimum length between two given points. Let u = I \/(l +> 2 ) dx ; then as far as terms of the second order inclusive, P B .. dx , r^ + 2 Thus in the usual way we obtain and this corresponds to a minimum. As long as we impose no other condition there will be no maximum. But now let us propose to find a line of maximum length between two fixed points with the condition that y is always posi- tive and that never changes sign. DISCONTINUITY PRODUCED BY CONDITIONS. 39 Let A and B be the two fixed points ; take A for the origin, .7) and draw BG and BD perpendiculars on the axes. Then the con- ditions imposed assign AD and DB as forming one boundary which the required line must not transgress, and A C and CB as another boundary. The preceding investigation shews that there will be no maxi- mum whatever so long as we take any line except one of these imposed boundaries. 41. It remains to investigate whether these boundaries them- selves are the required lines of maximum length. In order to avoid infinite values of p we will transform to polar co-ordinates ; the origin may be supposed to be at (7. r'Sr Let u = I fj -jr 8 + ( JQ\ [ dQ ; then to the first order f* f r < l r \ ,, 8 . /X - J j LV( r + r ) where r' is put for ^. Now the expression is zero both . 2 J(r +r 2 ) along AD and along DB ; thus the part of Su which is under the integral sign vanishes. 40 DISCONTINUITY PRODUCED BY CONDITIONS. The other part of Bu does not vanish. For the straight line AD we have r cos , where A C = a : thus -Tgr = sin 0. Simi- larly for the straight line DB we have r thus -TT-T- TUT = - cos B. , where = b : B Hence finally Bu = (sin DC A + cos D CA) Br where Br corresponds to the point D. And as when we pass by variation from the boundary ADS to an adjacent curve we must keep within the boundary, Br is essentially negative. Thus Bu is necessarily negative and our result is a maximum. [If however Br corresponding to the point D is zero, then Bu is of the second order instead of the first order, and is positive ; but di/ then ~- does not retain the same sign throughout.] In like manner the boundary A CB constitutes a maximum. 42. It is required to draw a curve of given length between two fixed points so that the area bounded by the curve and the straight line joining the fixed points may be a maximum. This is a well-known problem ; the curve must be an arc of a circle : see Todhunter's History of the Calculus of Variations, page 69. But suppose we impose the condition that the curve is DISCONTINUITY PRODUCED BY CONDITIONS. 41 not to pass beyond a certain given straight line which contains one of the fixed points. Let A and B be the fixed points, and OA a fixed straight line which the required curve is not to transgress. Take any point in this fixed straight line as the origin of polar co-ordinates. As the curve is not to transgress the fixed straight line, the most general supposition we can make is that it must consist of some portion A C of this fixed straight line, of length at present un- known, and of some curve CB. Let AC=r Q , and let the angle BOA = /3. Then we have to find the maximum of ^ I r*d0 while r + I given value. Hence a denoting a constant, we have by the usual theory to seek the maximum of cWo Denote this by u. Then in the usual way we make the part of $u which is under the integral sign vanish : and thus we find that the curve must be a circular arc. Then we have left 42 DISCONTINUITY PRODUCED BY CONDITIONS. To make this vanish we must have l- infinite, that is / f T-l =0; thus the circular arc must touch the fixed straight line. Then r must be determined from this condition, and the cir- cumstance that the whole perimeter is a given quantity. 43. For a more general problem we may take instead of a fixed straight line passing through A a fixed curve; and impose the condition that the required curve shall not pass beyond this fixed curve. It is natural to conjecture that if the given perimeter is so large that the required curve consists in part of an arc of the fixed curve, then the circular arc will touch the fixed curve at the point of junction. This we shall now verify. Suppose the required curve to consist of AP a portion of the fixed curve A PC, and the arc PB which we know will be an arc of a circle. Take any point in AB as origin of polar co-ordi- nates. Let the unknown angle A OP be denoted by 7. Put I for AP and S for the area A OP. Then a denoting a constant, by the o usual theory we must investigate the maximum of al + S + f" ^ + a V(r 2 + r' 2 )l dO. Denote this by u. Then by considering the part of Su which DISCONTINUITY PRODUCED BY CONDITIONS. 43 is under the integral sign and making it vanish, we obtain a cir- cular arc for the curve PR Thus we have left where for 6 we are to put 7. Now suppose that r = ^r(6} is the equation to the fixed curve ; then $1 = Vr^T}^' (0)) 2 dO. Also SS = dO. Thus Su = a Ji* + W (0)}' d0-aW+ O M - where for 6 we are to put 7. But since the point P must be on the fixed curve we obtain [by a process like that in Todhunter's Integral Calculus, third edition, Art. 359J the condition &r = W(ff)-r'}d0; and thus Hence that this may vanish we must have when 6 = 7 x this leads to {/ - ^'(W = > so tnat ^ = -^'(0). Thus the statement is established. 44. In like manner we may treat the problem in which there is also a fixed curve passing through B which the required curve must not transgress. As an example we may refer to the special case originally discussed by Legendre where the boundaries which the required curve must not transgress consist of two parallel straight lines, one passing through one of the fixed points, and the other through the other fixed point. 44? DISCONTINUITY PKODUCED BY CONDITIONS. 45. An area is to be bounded by a perimeter of given length ; the perimeter is to be constrained to pass through a certain number of fixed points : determine the form of the figure so that the area may be a maximum. [This problem was proposed by the present writer in the Mathematical Tripos Examination of 1865.] Suppose for simplicity of conception that there are three fixed points. Take an origin of polar co-ordinates within the triangle formed by joining the points. Then a denoting a constant we have by the usual theory to find the maximum of with the condition that for three assigned values of 6 the values of r must be equal to certain known quantities. Hence we shall obtain for the required curve three arcs of circles all having the same radius, namely a. Thus we have the discontinuity of a curve composed of arcs which meet at a finite angle. 46. But now let us advert to the case in which the given perimeter is so long that the arcs of circles would intersect outside the triangle formed by joining the three fixed points. In this case some portions of the area would in fact be counted twice over. If however we reject this as inconsistent with the nature of the problem, we must consider what modifications we have to make in our solution. From the discussion in Art. 42 we are now led to the following conclusion : Let A, B, C be the given points ; then the required figure will be composed of certain straight lines AA, BE', CC', and of certain circular arcs A'B' y B'C', C'A', all having the same radius. Moreover the circular arcs will touch the straight lines which they respectively meet. DISCONTINUITY PRODUCED BY CONDITIONS. 45 47. We may give a mechanical aspect to this problem. Suppose a cylindrical vessel to be formed of flexible material, and placed on a horizontal plane ; and suppose that the material is constrained to pass round fixed vertical rods. Let fluid be poured in ; then we know that in the state of stable equilibrium the area of the base will have a maximum value, so that the centre of gravity may be as low as possible. We know from almost elementary considerations that each portion of the boundary of the base will be an arc of a circle. The fact that the radii are all equal may be deduced from the relation 46 DISCONTINUITY PRODUCED BY CONDITIONS. which connects the pressure and tension with the radius : the rela- tion is given in books on Hydrostatics. Then, if the perimeter of the flexible material is large enough, we obtain the result explained in Art. 46. The lengths of the straight portions AA', BB' t CG' will of course be counted twice. 48. The discussion of the lengths and the positions of the rectilinear parts of the figure is not a problem of the Calculus of Variations, but of ordinary Geometry and Differential Calculus : we will therefore offer only a few remarks on it. 49. We shall shew that as the perimeter is gradually in- creased a rectilineal portion occurs first at the largest angle of the triangle ABC. Suppose the arcs which have AB and A G as chords to touch at A. Let r denote the radius of each arc; and let a, b, c denote the sides of the triangle. Then A + cos* -^ + cos* ^ = IT .................... (1). If instead of touching at A the arcs which have B as a common point touched we should have (2). Now if A is greater than B> then cos" 1 =- is greater than cos" 1 - . This shews that as the perimeter is gradually increased the arcs touch first at the largest angle. [Because as r gradually increases we arrive at the value which corresponds to (1) before we arrive at the value which corresponds to (2).] DISCONTINUITY PRODUCED BY CONDITIONS. 47 50. Now suppose that there is only one rectilinear part of the figure, namely AA' t Let sin BA'A c_ sin CA'A b ~~^0~~BA" sin(27r-^-0)~CM' ; , sin .5^'^ CA' csin0 therefore sin CAA BA' ' b sin ffrr~A+ty' And 2r sin BAA = BA', 2r sin CA'A = CA', therefore This gives a curve of the third degree as the locus of A'. -r f /x sin 6 c f we put p = we get s _____ _, this determines the direction which A4' initially takes. If we denote by D the middle point of BO we shall find that initially O = TT- CAD. For denoting CAA' by < we have sin (TT - 0) sin (7 / T - sm (TT ) sin , and TT , sin CL4J) sin C , ^ . ^ T> >t ^ also rs 77x7^ = - ^ , and CAD + BAD = A. sin B 48 DISCONTINUITY PRODUCED BY CONDITIONS. 51. Next consider the case in which there are two rectilinear parts of the figure, namely AA and BB'. Thus two circular arcs described on CA and B 'A as chords with the same radius touch at A ; and similarly those described on CB' and B'A' touch at B'. It follows by applying such equations as (1) and (2) of Art. 49 to the triangle CAB' that CB' = CA. Let A A and B'B meet at 0. Then A0 B0\ for they are tangents to the same circle. Hence the angle CB'B = the angle CA'A. = , B'BA=0'. Now Similarly sin CA'A CA A , , - n A A'Tr/Fi therefore sin CAA CA sin CAA = -^-p sin (2?r A ff). sin CB'B = - sin (27T - B - &) ; L/JJ therefore CB CA that is _^_^=_ sin ^1 sm B DISCONTINUITY PRODUCED BY CONDITIONS. Also OA = OB' ; that is c sin c sin 0' , sin (0 + 0') ^ sin And as CA' = CB' we have 49 (2). ....... (3). The equations (1), (2), and (3) will theoretically furnish by elimination one relation between p and 0, and one relation between p and 6'. 52. In the case in which the perimeter is so large that there are three rectilinear parts of the figure, the result is very simple. Let A A', BB', CC' denote these rectilinear parts. The tri- angle A'B' C' will be equilateral. The three straight lines A' A, B'B, C'C being produced will meet at a point 0, so that the angles AOB, BOG, CO A will all be angles of 120. Thus will be a fixed point in the triangle AB C. If the triangle ABC has an angle greater than 120 suppose it to be the angle A\ then the fixed point will be outside the triangle ABC, and BOC will be an angle of 120 while BOA and CO A will each be an angle of 60. 4 50 DISCONTINUITY PRODUCED BY CONDITIONS. 53. Suppose it required to find the greatest area included within a given figure, and having a perimeter of given length. For simplicity let us take the given figure to be a rectangle ABCD ; let AD be the shorter side. I. If the length of the given perimeter does not exceed IT AD the required solution is of course a circle. II. If the length of the given perimeter exceeds IT AD the required solution cannot be a circle ; for a circle with a perimeter of the given length would not fall entirely within the rectangle. But we are sure that the perimeter cannot consist of anything but some combination of circular arcs with parts of the boundary of the rectangle. For as in Art. 17 we know that in general it is necessary that the part of the variation under the integral sign must vanish, the only exception being when, as at the boundary of the rectangle, the quantity denoted by Sy cannot take either sign. Hence in the present problem the required solution must con- sist of a combination of straight lines and arcs of circles. By considerations similar to those in Art. 42 we infer that the straight lines will touch the arcs. Thus we obtain two straight lines EF DISCONTINUITY PRODUCED BY CONDITIONS. 51 and GH, and two semicircular arcs FG and HE\ the lengths of EF and GH being of course determined by the condition that the whole perimeter shall have a given length. This solution holds provided the given length does not exceed III. If the given length exceeds IT AD + 2AB - 2 AD the re- quired solution consists of four straight lines, EF, GH, KL, MN M connected by four quadrants of a circle, FG, HK, LM, NE, [of the same radius]. 54. Required to find the shortest path from a fixed point A to a fixed point B, supposing a circular obstacle having its centre in the straight line AB ; and supposing that the path is never to be convex towards AB and to have its radius of curvature never less than a given quantity r, and to have no abrupt change of direction. I, If the radius of the obstacle is not less than r the path consists of two tangents to the obstacle with an arc of the obstacle. 42 52 DISCONTINUITY PRODUCED BY CONDITIONS. II. If the radius of the obstacle is less than r the path con- sists of an arc of radius r which touches the obstacle, and straight lines from A and B which are tangents to this arc. If the obstacle is midway between A and B the two tangents will be of equal length, the arc touching the obstacle at the point most distant from A B ; but if the obstacle is not midway between A and B the point of contact of the arc and the obstacle must be found by the Differential Calculus. III. The solution holds so long as the arc which touches the obstacle cuts AB at two points between A and B. If this cannot be secured we must describe a circle on AB as chord with radius r: then if the shorter arc will not clear the obstacle the longer arc must be taken. If however we are not to transgress the limits obtained by drawing straight lines through A and B at right angles to AB, it will in this case be impossible to find any path that satisfies the conditions, and so of course there can be no shortest path. 55. It is easy to justify the statements in the preceding Article by the Calculus of Variations. Let u = I >v/(l -f p 2 ) dx] so that when the integral is taken between proper limits u denotes the length of the path. Then to the second order inclusive DISCONTINUITY PRODUCED BY CONDITIONS. 53 The term outside the integral sign vanishes since the extreme points A and B are fixed. Thus we have Now &u is positive. For along the straight lines we have #"=0, so that &u reduces to the second part of the above expres- sion which is a positive term of the second order. And along the circular arc we have y" negative, and it will [probably] be found that we cannot suppose By negative without breaking the con- ditions which have been imposed, that the curve is not to be convex to the axis, and that the radius of curvature is not to be less than r, and that there is to be no abrupt change of direction. Thus we see that we have a minimum, and we infer that it is the least value because no other presents itself. 56. We have thus briefly considered various simple examples of discontinuous solutions ; we shall now proceed to a full dis- cussion of some problems of historical interest in the Calculus of Variations, which involve discontinuity. CHAPTER IV. MINIMUM SUEFACE OF REVOLUTION. 57. REQUIRED the plane curve joining two given points which by revolving round a given axis in its plane will generate a surface of minimum area. This problem has been much discussed : see the prize essay by Goldschmidt noticed in Todhunter's History of the Calculus of Variations, page 340 ; see also Professor Jellett's Calculus of Variations, 1850, page 145, the Calcul des Variations, published in 1861 by Moigno and Lindelof, page 204, and Dienger's Grundriss der Variationsrechnung, 1867, page 15. I shall add something to the researches of previous writers. 58. Take the axis of x as that of revolution. Then we require the minimum of \y *J(\ +p*) dx y the limiting values of x and y being fixed. We obtain in the usual way Vfl+jp*) = l a constant ; thus jP 2 =^^- 2 ; , dx a therefore - therefore x+ 0, = ^lo MINIMUM SURFACE OF REVOLUTION. 55 Taking either sign we have the equation to a catenary of which the axis of x is the directrix. We have then to examine this solution. 59. First we ask if it is possible to draw a catenary having a given directrix, and passing through two given points. This question has been considered by previous writers ; the conclusion is that sometimes two catenaries can be drawn, sometimes only one, and sometimes none. I shall however presently give a new investigation of the question, simpler I believe than those hitherto published. The next question is to determine whether we really obtain a minimum. I shall shew that when two catenaries can be drawn the upper corresponds to a minimum, and the lower does not; and that when only one catenary can be drawn it does not cor- respond to a minimum. These statements are new. Goldschmidt erroneously thought that both catenaries corresponded to a mini- mum : see his page 27. The other writers do not discriminate between the two catenaries, except Moigno and Lindelof in a particular case. Strauch is not very full on this problem; he considers that there is always a minimum corresponding to the catenary : see his Vol. II page 276. Stegmann does not discuss the problem : he barely alludes to it on his page 187. 60. It is convenient to begin with the particular case in which the given points are equally distant from the axis ; to this par- ticular case some previous writers have practically restricted them- selves. Let the distance of each given point from the axis be b ; and let 2a be the distance between the points. Then we assume for the equation to the catenary c x -- " so that c has to be found from the equation & = e? + e -? ................ ........... (1). 56 MINIMUM SURFACE OF REVOLUTION. We have then in fact to determine if the last equation gives a real value or real values for c. c - Denote ^ (& + e~^) by < (c) ; regard a as a constant and c as a variable. Then we see that (c) is infinite both when c = 0, and when c = oo . And From (2) we can shew that $ (c) vanishes once, and only once, as c ranges from zero to infinity. For by expanding we get .,,; la 2 3 a 4 2n-la* n * (C)= -2cMl '""" T^~ ?= "5 so that <>' (c) is negative infinity when c is zero, and is unity when c is infinite, and changes sign once, and only once, as c passes from zero to infinity. And (c) has its least value when <' (c) = 0. If then the given value of b be greater than the least value of < (c) there are two values of c which satisfy (1) ; if the given value of b be equal to the least value of (c) there is only one value of c ; if the given value of b be less than the least value of (f> (c) there is no possible value of c. It has been found that the value of - which makes c a _a n. a a e c + e ~c (e e~) = 0, is approximately - = 1-19968 ...; and then it follows from (1) c that - = 1-81017...; and therefore - = 1-5088...: see Dienger, c a and Moigno and Lindelof. Thus there are two catenaries satis- fying the prescribed conditions, or one, or none according as - is greater than, equal to, or less than 1-5088. 61. Now we pass to the question whether corresponding to a catenary the surface generated is a minimum. We know that MINIMUM SURFACE OF REVOLUTION. 57 to ensure a minimum the tangents to the catenary at the fixed points must intersect above the axis of x ; see Art. 29. From symmetry the two tangents in the present case will intersect on the axis of y. The equation to the tangent to the catenary at the point (a, b) is y-b=p(x-a\ 1 - -- where p = ~ (e c e c ) ; therefore the ordinate of the point where this crosses the axis of y is b pa, that is c a - - a ~ -- _(^ +e - C )--( 6 c_ e c). And it is obvious from our discussion of the value of cf> (c) that the above expression is positive for the larger value of c obtained from (1) and negative for the smaller value of c. Hence when two catenaries can be drawn the upper catenary corresponds to a minimum and the lower does not. When only one catenary can be drawn it does not correspond to a minimum. The result for this particular case had been obtained by Moigno and Lindelof : see their page 210. 62. We shall now discuss the general problem. Let b be the distance of one given point from the axis of x, and k the distance of the other. Let the axis of y be placed midway between the given points ; take a for the abscissa of the former given point, and a for the abscissa of the latter. Then we take for the equation to the catenary C where n and c have to be found from the equations ,(3). ^ a+n a+n -, 6-|( + e-~ ) c n-a n-a _ - ^ 58 MINIMUM SURFACE OF REVOLUTION. From (3) we obtain c n , 2a 2a a _a -. e c (ec e c ) = be c ke c .(4). C - -& c e c __ c And from (4) by multiplication c 2 / ?? _??\ 2 a _ a _ 7-( ec c ) = (be ke~ c ) (kec be ) .......... (5). Thus we have eliminated rc and obtained the equation (5) for determining c : we have now to examine if this equation gives a real value or real values for c. Let (c) stand for 2a g j v 7- \e c e"" ) (b& Jce'c) (Jcec be~ c ). * Then (c) is infinite when c is zero, and is 4ta* + (b &) 2 when c is infinite. We shall find that 'c e u ij W ^o-j. e J I 1~| 2a _2a The factor e c e c cannot change sign. The other factor of (c) becomes by expansion a If 5A; is not greater than then ' (c) never changes its sign ; and the least value of < (c) is when c is infinite : as this value is positive it follows that (c) cannot vanish. If bk is greater than 4a 2 then <' (c) changes its sign once, and only once ; so that 6 (c) has a corresponding minimum value, and according as this value is negative, zero, or positive we have from (5) two values of c, or one, or none. MINIMUM SURFACE OF REVOLUTION. 59 63. We now pass to the question whether corresponding to a catenary the surface generated is a minimum. As before we must determine whether the tangents to the catenary at the fixed points do or do not intersect above the axis of x. Let PI stand for the value of -~ at the point (a, b) ; and p^ for the value of ~ at the point ( a, k) : then the equations to the tangents are respectively y-b = p 1 (x-a), y-k =p 2 (x + a) ; at the point of intersection therefore y = y-k-p z a p + *P. - By using (4) we obtain for the equation to the catenary ^ f x a a x a _-) y = - \ec fy e c _ ke~~?) + e~ c (ke^ be ) K 2 2a where X stands for e~c e~ c . Hence we shall find that 2a _2a where /* stands for e c + e c . Thus Pi~p z = 1 - - which is positive. AC We have now to examine the sign of Zap^ + 7^ From the values of^ and^? 2 we obtain _ ty (b* + k*) - (4 + /) bk 60 Thus MINIMUM SURFACE OF REVOLUTION. 2 ~ A C {(b z + F - - Xc) But equation (5) may be written and so that 7 Xc - % = - ytta + Now from our investigation of the value of ' (c) in Art. 62 it follows that when there are two admissible values of c the expres- Xc , 2abk . . . f , sion - pa 4- is positive for the greater value of c and nega- _ c tive for the less ; and when there is only one admissible value this expression is zero. Hence when two catenaries can be drawn the upper corresponds to a minimum, and the lower does not. When only one catenary can be drawn it does not correspond to a minimum. We may remark that the two catenaries which present them- selves in this problem correspond to the figure which a uniform endless string will assume when hung over two pegs. 64. We shall now consider a discontinuous minimum which always exists. This has been noticed by no writer, I think, except B Goldschmidt ; he briefly adverts to it, but does not shew that it is a minimum. MINIMUM SURFACE OF REVOLUTION. 61 Let A and B be the given points ; A C and BD the perpen- diculars from them on the axis. Then the discontinuous solution is furnished by taking the generating curve to consist of A C, CD, and DB\ so that the surface consists of the two circles having CA and DB respectively for their radii, connected by the straight line CD : this connecting part may be conceived to be an infinitesimally slender cylinder. We shall hereafter consider the origin of this discontinuous solution: see Art. 68. 65. It is very easy to shew that the proposed solution really gives a surface of minimum area. Let the dotted line in the diagram represent a closely adjacent curve. Set off from A along A C and along the dotted line equal infinitesimal lengths. Let PQ and pq be a corresponding pair. Then it is plain that PQ will be rather nearer to the axis of revolution than pq is; and so pq will generate a somewhat larger element of area than PQ will. In like manner if we set off from B along BD and along the dotted line equal infinitesimal lengths we find that the element of the dotted line generates a somewhat larger element of area than the element of BD does. And CD generates no area, while any element of the dotted line in the neighbourhood of CD does generate an area since its distance from CD is not absolutely zero. Hence we see that the dotted line generates an area which is certainly greater than that of the proposed discontinuous solution ; in other words the discontinuous solution really is a minimum. 66. The same result may be obtained by the ordinary methods of the Calculus of Variations. In order to avoid infinite quantities we will use polar co-ordi- nates. Suppose the initial line parallel to DC ; let k be the dis- tance of the origin from D C ; and let the vectorial angle increase in the direction from A towards B. Then the integral we wish to make a minimum is 3B M ' ^au/ the limiting values of the variables being fixed. Denote this 62 MINIMUM SURFACE OF REVOLUTION. f\n4- IY\ rV\T* dd' . , , , ; f I 2 , (dr\* , f dr integral by w; and put v for */ r + [ ja) ', also put p for Then to the first order ff U /I . a* k , - v sin 0Sr H -- r or + This transforms into k rsin# -, ff . k rsin0 d (k r sin 6} p} ~ ^ v * Jj v dO v ) The term outside the integral sign vanishes, for Sr = at the points A and B ; and although Sr is not zero at the points C and D where we have discontinuity, yet at these points k r sin = 0. We have now to determine whether the term in Su, which is under the integral sign vanishes for every point of the discon- tinuous line which is under examination. Consider the part CD ; here Jc r sin 0, so that the co- efficient of Sr reduces to . * p (p sin#-l-rcos#) v sm 6 + F vr , v . , , . 4 pr cos r 2 sin that is to * . v Now when Tc = r sin we get p r cot 0, so that the co- fc 2 efficient of Sr is : ^ , which is a negative quantity and not v sm 6 zero ; but along CD we have Sr necessarily negative, since there is obviously the implied restriction that the generating curve shall be above the axis of a?, or, which is the same thing, that the surface generated shall be taken positively. Hence as Sr is negative &u is positive so far as the elements arising in connexion with CD are concerned. Next consider the part AC\ here we have rcos# = a con- stant = I say. This gives p - ^-^ , v = 5-3 . Thus the co- J f cos 2 6 cos* 6 MINIMUM SURFACE OF REVOLUTION". 63 efficient of Sr becomes I sin 6 7 Q 7 - a d (k cos 6 I sin 6} sin 6 -- o-/i + & cos Z sm 6 -j-~ - - ^ ' cos 2 d# cos0 sin d 1(1 -cos 2 0) - - = 0. -TVl V OJ.L1 V -f- -.ft ft cos 2 c?0 cos Similarly &u is zero so far as the elements arising in connexion with BD are concerned. Thus on the whole we have Bu a positive quantity of the first order, and so the proposed solution is really a minimum. 67. Thus so long as we consider a curve which is adjacent to the discontinuous line but which differs from it through the whole extent, we are sure of a minimum without examining the terms of the second order in the variation. But if we do not vary the part CD our conclusion that Su is a positive quantity of the first order does not hold : so that we are interested in examining the terms of the second order in $u. These terms consist of - 2p sin _ 2(k-r sin 0) rp ~ ~ v v 3 ~ " P The second of these two terms is never negative. We will transform the first term. We have f J sin ~ . j- 2 p sin 9 L d (p sin 6 8r\ - cr &p dd = (Sr) 2 *- --- \9r-sA I ~ - v * ^ ' v j d6\ v J 64 MINIMUM S DEFACE OF REVOLUTION. therefore _ f p sin ~ 5, JQ /S s N2 P sin f.^ > 2 d fp si JQ 2 J ttrtpM- (r) 8J -^- - - Thus the first term becomes TVT f A n ^ . r sin 6 1 d p sin d , .Now for AC the expression 9,~Jf) reduces to sin 6 cos 6 ^ - sin 2 6, that is to zero ; and similarly for BD it also reduces to zero. For CD this expression reduces to sin 2 + JQ cos 6 sin 0, that is to ^ . The term outside the integral sign, namely, (Sr) 2 *= , will not vanish at C and at D, for - has two values at C and D by v J reason of the discontinuity, namely, * = sin 6 along J. (7, and - = cos along CD, and then ^ = sin 6 along D.Z?, that is from J) towards B. Let r l and t be the co-ordinates of (7, and r 2 and 2 the co- ordinates of D. Then finally we get for the terms of the second order in &u (Sfj) 2 sin l (sin X + cos 0J ^ (Sr 2 ) 2 sin 2 (sin 2 cos 2 ) ^ where the last integral extends over the whole discontinuous line from A to B. MINIMUM SURFACE OF REVOLUTION. 65 Now we cannot assert that this expression of the second order is always positive ; but we do not require that it should be so : for by the preceding Article all we require is that this expression should be positive when Sr is supposed to vanish along CD. In this case r l = 0, Sr 2 = 0, and I (Sr)V0=0; so that the expres- J 0! sion is positive. 68. The question may be asked whence does this discon- tinuous solution arise ? The reply is that the part CD presents itself in accordance with the general principle of Art. 17; for along that line 8y is not susceptible of either sign. The two parts A C and ED are implicitly involved in the fundamental differential equation of Art. 58, namely -= constant inasmuch as^p equal to infinity, combined with the constant equal to zero, may be considered as a solution of the equation. 69. The conclusion of the investigation is as follows : the problem enunciated in Art. 57 always admits of a certain discon- tinuous solution, and sometimes admits of a certain continuous solution. When both solutions are admissible we shall find that sometimes the discontinuous solution is the less of the two, and sometimes the continuous solution. If the two points are very near each other and at a great distance from the axis of x it is plain that the proper catenary will give a less surface than the discontinuous solution. If the given points are so situated that the two catenaries nearly coincide, the discontinuous solution gives a less surface than the proper catenary. For let 8 denote the surface generated by a portion of a catenary extending from the lowest point to any point P et the tangent to the catenary at P meet the axis of x at a point T the abscissa of which is f . Let S denote the surface generated by the revolution of PT round the axis of x. Then it is known that 66 MINIMUM SURFACE OF REVOLUTION. being considered positive or negative according as P and T are on the same side of the axis of y, which is supposed to pass through the lowest point of the catenary, or on opposite sides of it. See Moigno and Lindelof, page 212. Hence it will follow that the entire surface generated by the revolution of an arc of a catenary round the axis of x is greater than, equal to, or less than that generated by the extreme tangents according as the intersection of these tangents is below, or on, or above the axis of x. The surface generated by the tangents is of course greater than that generated by the extreme ordinates, that is greater than the discontinuous minimum surface. If the numerical values of the extreme ordinates are given in any case, as well as their distance apart, we can by numerical calculation find the approximate value of c if there be a possible value ; then calculate the surface : and so determine whether the continuous minimum is greater or less than the discontinuous minimum. 70. We have spoken throughout of a minimum surface. It is sufficiently obvious however that there must be a least surface ; and as no other solution can be found there is no doubt that the least surface is one of the two minimum surfaces when both these exist ; and when the discontinuous minimum is the only one that exists it is the least surface. 71. The preceding problem gives a simple natural illustration of the general principles which we have laid down : see the re- mark III. of Art. 14 and also Art. 18. 72. An important part of the preceding investigation con- sists in shewing that when the two catenaries coincide the tan- gents at the fixed points intersect on the directrix. We may easily establish this result independently. Let the equation to the catenaiy be MINIMUM SURFACE OF REVOLUTION. 67 let the abscissae of the fixed points be a and h, and the correspond- ing ordinates b and k: so that fi+n ,(i). We require that (1) should be true also when c and n receive indefinitely small increments Be and Bn. Put i/r (x) for -~ , that ax J x+n _ 2 ( e ~^ ~ e ~ ~ 7 1- Then from (1) 6Sc - (a 4- ?i) ^ (a) Be - c^r (a) Bn = 0, Be - c^ (k) Sn = 0; 5 - (a + n) & (a) i/r (a) therefore -j 77 - , ; 7 ; = r ~~ : k-(h + n)^ (h) ^ (h) ' therefore ^ (A) [a^ (a) - b} = ^ (a) [^ (A) - Jc} ............ (2). But the equation to the tangent to the catenary at the point (*#) is y l -y = ^r(x) (ajj - x\ where x l and y t are the variable coordinates. Therefore at the intersection of the extreme tangents we have y l - b + a^ (a) jr (a) In order then that y^ may be zero it is necessary and sufficient that t (h) [a^r (a) - b} = ^ (a) {h^ (h) - k] ; and this agrees with (2). Thus the required result is established. 52 CHAPTER V. MAXIMUM SOLID OF REVOLUTION. 73. To determine a solid of revolution the surface of which is given, so that it may cut the axis of revolution at given points and have a maximum volume. This problem has given rise to some discussion and contro- versy, as will be seen by consulting the volumes of the Philoso- phical Magazine for 1866. I shall repeat with brevity what has already been established with respect to the problem, and then proceed to additional investigations. Adopting the usual notation we have to make TT I y z dx a maxi- mum while 2-7T I y V(l +p 2 ) d& is given ; the limiting values of the variables being fixed. Thus by the usual method we require the maximum of where a is a constant at present undetermined. Denote the in tegral by u ; then to the first order where M stands for MAXIMUM SOLID OF REVOLUTION. 69 By the known principles of the subject we put Jf=0, and this leads in the usual way to Since the generating curve is to meet the axis of x we have y = at certain points ; hence b = Q, and the equation just ob- tained becomes 7(1 Thus we appear to have either :. r + y or y = 0. If we take /n ^ 2N -f y = 0, we obtain vi 1 therefore -j- dy This gives us a circle of radius 2a having its centre on the axis of x. But if a circle has its centre on the axis of x and passes through two fixed points its radius is determined ; and so the corresponding surface of the sphere cannot have a given value. Thus we have not a satisfactory solution of the problem. This led to the suggestion by the Astronomer Royal that the solution of the problem is to be obtained by combining the two results ; and taking -^- ' ^ + y = for part of the required line, and y = for part of it. This gives for the solution a sphere which is connected by a straight line, namely part of the axis of revolution, with the fixed points. For facility of conception we may consider this straight line as an infinitesimally slender cylinder. 70 MAXIMUM SOLID OF REVOLUTION. 74. But on examining the proposed solution we find that the supposition y = does not make M vanish ; and thus at first sight the proposed solution appears unsatisfactory. Nevertheless there is no doubt that we have the true solution here ; the apparent difficulty is removed by the principle of Art. 18. For corresponding to y= the value of by is essentially positive ; hence we are not compelled to have Jf=0: it will be sufficient that M be negative. Now when y=Q we have Jf=2a, and 2a is necessarily negative, as we see from the equation which holds when y is not zero. Thus in fact instead of having &u = so far as the first order of small quantities, we have &u a negative quantity of the first order : and therefore a maximum is ensured. [This remark, which is essential to render the proposed solu- tion admissible, was supplied by the present writer ; it was the first introduction of the important principle of Article 18 into the Calculus of Variations.] 75. Thus as long as we consider a curve which is adjacent to the discontinuous line but which differs from it through the whole extent, we are sure of a maximum. But if we do not vary the part which corresponds to y = 0, we should have to appeal to some other evidence to shew that we have secured a maximum. It is however unnecessary to investigate the terms of the second order in $u ; for we may rely on a theorem which is well known, that the sphere is the body which has the greatest volume under a given surface. 76. The following then is the result : Let A and B denote the two fixed points on the axis of revo- lution ; then according as the given surface is greater or less than that of the sphere having AB as diameter, we have the upper diagram or the lower diagram. The generating curve must be supposed to be made up of the two rectilinear portions A C and MAXIMUM SOLID OF REVOLUTION. 71 BE and the semicircle ODE. In the particular case in which the given surface is equal to that of the sphere having AB as diameter the rectilinear portions disappear. The result may appear strange, at least to any person who was not familiar with the considerations brought .forward in the preceding chapters of these researches : we will make a few re- marks on the result. It is certain that the sphere is the solid of greatest volume within a given surface ; and thus, admitting that our solution does fulfil the prescribed conditions, we are certain that it is the greatest solid which will do so. But an objector might say that he wants the greatest solid with the condition that the generating curve shall have no abrupt change in direction, and shall cut the axis instead of partly coin- ciding with it. I reply that nothing can be obtained which differs to an appreciable extent from the solution already given. It is obvious that we can draw a curve fulfilling the two conditions thus 72 MAXIMUM SOLID OF REVOLUTION. stated and deviating innnitesimally from the straight lines and semicircle ; so that the surface and the volume will differ only to an infinitesimal extent from those of our solution : or if we make the surfaces equal the volumes will differ only infinitesimally. See the remark III. of Art. 14. A good method for drawing these curves theoretically would be to employ the propositions which serve as the foundation for the expansion of functions in terms of sines and cosines of multiple angles. 77. Abandoning then the attempt to obtain any other solution for the greatest solid than that which we have given, the objector may still say that he asks for a maximum solid among all those which have a given surface, the generating curve being constrained to cut the axis and to have no abrupt change of direction. I say that it is hopeless to seek for any such maximum distinct from what we have given. For there being no restriction introduced as to the sign of By no one will hesitate to admit that the condition which we denote by M must be satisfied ; and this necessarily leads to for the equation M = when developed is x _ = dp n zayp -~- that is 2y + -.*. -- -SL = ; and as this is to be true for all values of y we may integrate with respect to y : and thus we have Thus we cannot avoid arriving at this equation ; and then we must continue the investigation as in Arts. 73 and 74. MAXIMUM SOLID OF REVOLUTION. 73 78. We have assumed in Art. 76 that when the given surface is greater than that of a sphere having AB as diameter, the solid may stretch beyond the straight lines at right angles to the axis at A and B. If however the solid is restricted to lie between these straight lines, the solution is that given in Todhunter's History of the Calculus of Variations, page 410. 79. Although the problem enunciated in Art. 73 does not admit of solution except in the way we have explained, yet con- ditions may be introduced which modify the problem, and so lead to other solutions. For example, let us impose the condition that the generating curve shall never be convex to the axis of revolu- tion, in addition to the former conditions of cutting the axis at two given points, and having no abrupt change of direction. Sim- ple as the problem still is in enunciation it does not very obviously appear what the solution will be ; and the remarks now about to be made will be confirmed or corrected if readers of the present researches will investigate the problem for themselves before con- sulting the solution which we shall now propose. 80. It will be convenient to change the enunciation of the problem to the following, which is of course substantially equiva- lent. To determine a solid of revolution of minimum surface, the volume being given ; supposing that the generating curve cuts the axis at two given points, that it has no abrupt change of direction, and that it is never convex to the axis of revolution. 81. If the given volume is that of a sphere on the intercepted portion of the axis as diameter the required surface is that of this sphere ; if the given volume is greater than that of this sphere the solution is that given in Todhunter's History of the Calculus of Variations, page 410. We have then to consider only the case in which the given volume is less than that of a sphere on the inter- cepted portion of the axis as diameter. 82. By the Calculus of Variations every kind of boundary of the generating figure is excluded except straight lines, and curves which satisfy the differential equation Jlf=0. At first sight it 74 MAXIMUM SOLID OF REVOLUTION. might appear that every line is excluded which does not satisfy the differential equation M = : but on consideration we shall see that straight lines are not excluded, because it may be possible that for straight lines Sy is not susceptible of either sign by reason of the condition which forbids convexity towards the axis. Moreover the curves which satisfy the differential equation M do not cut the axis ; excluding the particular case of a semi- circle which is here inadmissible : hence we are driven to the con- clusion that the portion of the required boundary in the vicinity of the axis must be rectilinear. 83. I propose the following for the solution of the problem : Let A and B be the fixed points on the axis ; let AP and BQ D be equal straight lines equally inclined to the axis which touch the curve PQD at P and Q respectively; and let PDQ be an arc of the curve defined by the differential equation . ,* , r . y " where a and c are constants. The constants must be taken so as to ensure the tangency at P and Q, and to make the volume gene- rated by APDQB have the given value. Moreover there is a certain condition to be satisfied which we shall investigate pre- sently ; this condition connects the constant a with the ordinate of P and Q, and the inclination of PA and QB to AB. This condition is y^ = -~- cos /3, where y l is the ordinate of P or of Q, and ft is the angle of inclination of AP or BQ to the axis AB: see Art. 85. MAXIMUM SOLID OF REVOLUTION. 75 It will be observed that the constant c 2 corresponds to the b of Art. 73, and that the present constant a corresponds to the a of that Article. 84. We proceed to shew that in the way just stated we do obtain a minimum value of the surface. It will be seen that the differential equation for determining PDQ is a first integral of the equation M = 0. Let 8 denote the surface generated ; so that then to the first order, ~ a %7n/pBy f( . 2 . d yp B8 = . ' yr \ + 2?r U/(l + p) - -j- -77/7- x/1 ' ) J a^c ;/1 ' 1 W } rs 7 5yfa?, ;-+/) (i where ^ stands for -~ . Both parts of the expression for B8 are of course to be taken between limits. Now by means of the equa- tion to PD Q we find that the coefficient of By under the integral sign reduces to - for the part PDQ of the boundary. For the a rectilinear parts q = 0, and so the coefficient of By under the inte- gral sign reduces to The term in 88 which is outside the integral sign vanishes ; because A and B are fixed points, and atP and Q the straight lines touch the curve. Thus BS consists of - - ly&ydx for limits corresponding to PDQ - r\ 7 and of 2-7T I-T ^ for limits corresponding to^tP and BQ ; that 76 MAXIMUM SOLID OF REVOLUTION. 2_ r is we may say that SS consists of - - I ySy dx for limits correspond- ed J ing to the whole line together with 2?r I \ - - ^ - \ &/ dx for J (V(l + p") a ) limits corresponding to ^4Pand BQ. Now since the volume of the solid is given we have so that to the first order of small quantities 2?r lySydx = 0. Thus finally BS reduces to 2?r I] ^ -[ Su dx over limits corre- J (V(l + p ) a ) ' sponding to AP and BQ ; and in order to ensure a minimum it is therefore essential that this expression should be zero or positive. 85. Let /3 denote the angle between AB and AP ; then along AP we have -j-= 27 = cos/3. Let y l denote the ordinate of P. Suppose y l such that y? cos P y? _ A 2 3a~ ,-, Set .^ so that y^ cos p. Then it is obvious that if we take Sy proportional to y we have I (cos /8 - j By dx = 0, for limits corresponding to A P. For if we put &y = //#, where yu, is a constant, the value of the integral Cii 2 cos Q 11 3 \ 9 ^ J ' that is zero. And we shall now shew that the integral will be positive for any other supposition respecting by consistent with the condition that there is no convexity. Throughout when we speak of taking By proportional to y, we mean that it is so along AP or BQ\ for other parts of the boundary y may be taken as we please. MAXIMUM SOLID OF REVOLUTION. 77 For let ALIK be a curve obtained from the straight line AP by ascribing admissible values to Sy: so that the curve has no convexity towards the axis of revolution, and no abrupt change of direction. 2 Let AF= g AP. Draw FH parallel to the axis of y to meet the curve at H, and draw the straight line AHG. Now if by relate to the straight line AHCr we have 8y pro- portional to y\ and then /(cos/3 -1 By dx for the limits with which we are concerned is zero ; that is I ( -~ yjSydx is zero. Hence we shall find that if By relate to the curve ALHK we must have a positive value for I (~^ yj 8y dx. For now there is a gain of positive elements corresponding to the area between AH and ALH\ there is a diminution of nega- tive elements in having HIF instead of HFPG, that is a relief from the negative elements corresponding to the area HIPGrj and there is a gain of positive elements corresponding to the area IKP. 78 MAXIMUM SOLID OF REVOLUTION. The preceding diagram supposes that FH has to be. drawn up- wards to meet the curve ; if FH has to be drawn downwards we have such diagrams as the following : In both cases the integral I ( -^ y}Sydxis zero when By re- J \ o / lates to the straight line A G ; and hence we shall find that the integral is positive when By relates to the curve AHK. For in the upper diagram there is a gain of positive ele- ments corresponding to the area ALT, a gain of positive elements corresponding to the area HKG, and a relief from the negative elements corresponding to the area AIH. In the lower diagram there is a gain of positive elements corresponding to the area HKG, and a relief from the negative elements corresponding to the area ALH. MAXIMUM SOLID OF REVOLUTION. 79 86. Thus we shew that for the line APBQ we have to the first order BS always positive except in one particular case, and then SS is zero. Such a result is as much as can be obtained in most problems of the Calculus of Variations ; although strictly speaking in the case in which SS is zero to the first order we ought to examine the terms of the second order to be absolutely certain of a minimum. I shall return to this point at the end of the present chapter. Of course even when we have secured a minimum the result is not necessarily the least which is possible ; but in the present case I think there can be little doubt that our result is really the least. There must obviously be some least value ; and the boundary must be composed of a straight line or straight lines and arcs of tho curve determined by M = ; and I believe that with due consider- ation every combination of these admissible elements, except that which we have adopted, will be excluded. 87. Let us now examine more closely the equation Sa y, = -g cos 0, which we have obtained for determining the points P and Q. It is well known that the equation belongs to the curve traced out by the focus of an ellipse as the ellipse rolls on the axis of x. See a memoir by Lindelof in the Acta Soc. Sri. Fenn., Helsingfors, 1863. The major axis of the rolling ellipse is 2a, and c 2 = a" (1 e 2 ), where e is the eccentricity. The curve thus generated is partly concave towards the axis of x t and partly convex. The distance from the axis of the highest point is a (1 + e). 80 MAXIMUM SOLID OF REVOLUTION. We are here concerned with the concave portion. If p be the radius of curvature at any point of the concave portion we have Thus at the points P and Q since the curve touches the straight lines we have &+.&.& 3a p a' therefore p = 3. Now the radius of curvature at the point of the curve which is most distant from the axis of x will be found to be - -- ; and e at the point of inflexion it is of course infinite. Thus we require that the value Ba should lie between - -- - and infinity ; that Q is e must be greater than ^ . JU We may determine the value of y^. Substituting in the equation we obtain = y* + a * (1 - e 2 ) ; o therefore y, 2 = 3a 2 (1 - e 2 ). If e = 2 we have y l = ~ , and therefore cos j3 = 1. Thus when e = 2*hQ straight lines AP and BQ coincide with the axis of revolution, so that the volume and the surface vanish. This is consistent with what we have already found, namely that - is a limiting value of e. MAXIMUM SOLID OF REVOLUTION. 81 88. Hence the following is our process of solution. Take a curve denned by the equation ^ 2 = y* + a? (1 - e 2 ), and draw tangents at the points for which the radius of curvature is 3a ; then we have to fulfil the following conditions : these tangents must intersect the axis of x at given points, and the solid gene- rated by the revolution of the figure round the axis of x must have a given volume. The conditions must serve for determining the constants a and e, as well as the constant which may be conceived to arise from integrating the differential equation to the curve. But practically we shall not be obliged to pay any regard to the constant introduced by integrating, since we may suppose the origin to be at any point we please in the axis of revolution. 89. Although we have now carried the solution as far as such solutions are usually carried, yet we will continue the investigation and shew that the conditions by which a and e are to be deter- mined can always be satisfied. As the relation between x and y for the curve PD Q cannot be exhibited explicitly, we have to adopt an indirect method. Let AB = Zh\ then ra(l+e) J y h = y l cotj3+ - , J Vi P I intend to shew from this equation, in which h is a fixed quantity, that a and e increase together. Let Then 2 V denotes the volume of the solid generated by the revolution of our proposed boundary. I intend to shew that as a and e increase V continually increases, so that we can make our volume equal to any assigned volume lying between zero and the volume of the sphere having 2h for diameter. 90. We have y cos 0, and cos 2 = - (1 - 82 MAXIMUM SOLID OF REVOLUTION, As p = when y = a (1 +e) we shall find it convenient to change the independent variable from y to p in our integrals. From the equation the upper sign applies to the part of the curve with which we are concerned: thus 1 + Therefore z 3acos 2 /3 Let J^ denote the expression under the integral sign ; and let Xj be the value of L when p = tan /3. Let a and Se denote simul- taneous indefinitely small changes in a and e consistent with the relation just expressed, in which h is given. We obtain then 3a d cos 2 /? ,- d \ d/3 ^--^ r S - 2 JS sin I shall shew that the coefficient of $e in this equation is nega- tive. It is obvious that -7- is negative. / S = - /3 \ dfi sin ft \ sin* p) ' It will be found that L v = 3 cos 3 ft. JO Thus the term involving ~ $e is MAXIMUM SOLID OF REVOLUTION. 83 ., , . 3a cos 3 /5 dp* that is, -g- ;-:-TID 77- $ e - Hence finally A ~ (3& cos s /5 e?/5 ) ~ - oa \-fi- . 2 Q -j- + XJ- o>, a 1 2 sin 2 /5 efe j where X stands for the positive quantity And -r- is positive for de sin P cos $ -~ = -= ; so that Sa and Se have the same sign. 91. Now we have V = 5- ; 5- + TTft I g . dp. 8 sm p J (1 +j9 2 )s ^/g 2 (l 4.^2) _p 2 Let J7 denote the expression under the integral sign, and let H t be the value of H when p tan/5. Then siipposing a and e to receive infinitesimal changes, and denoting by 8 V the consequent change in V, we have 3F, /97m 8 2 is less than 1 3 , that is less than ^ . dV . Hence j is -positive. da The above demonstration is substituted for that which was originally offered. In the original demonstration a property of a 2 (e) was employed, and as this property may be of use it will be given in the next Article. The reader who wishes to avoid an interruption to the reasoning may pass on to Art. 94.] 37r(3sin 2 /3-l)cos 2 /3 2^ no ITT i , \ sin 3 $ ea 3 93. We have a' (e) = 4 cos 2X 86 MAXIMUM SOLID OF REVOLUTION. We shall find that as e increases the denominator continually diminishes, and the numerator continually increases until sin 2 /3 is greater than ^ . o First consider the denominator. j. f tan ^ _ i_ __i _ ~ a* J V(l +/) X e 2 1 + 2 -* T _ __ ea* ~ a* V(l +/) {e 2 (1 + p 2 ) - _, 4cos 2 /3 2X Put .# for 2 . 30 + 3 . a sin /8 ea 7 r> Then -= consists of various terms every one of which is neces- da sarily negative, except that which arises from the variation of the upper limit in the integral involved in the value of X ; the term dR 1.1,1 in -f- which thus arises is da 2 R 1 1 dJ3 de de da' AU A . 16 d@ de that is -5 -Q -T- -T- a cos p de da j-p But -j- besides other negative terms has da 4 d_ /cos 2 /3\ d/3 de_ a? d@ (sin* 0) de da ' 8 cos 13 sin 2 + 12 cos 3 /3 J/3 ^e that is, 2 . 4 Q -j- -j- ; a 2 sm 4 ^ de da and this negative term more than counterbalances the positive term just given; for (8 cos/3 sin 2 /3 + 12 cos 3 ft) cos/3 is greater than 16 sin 4 0, provided sin 2 /? is less than ^. To shew this we o must compare cos 2 (2 + cos 2 /3) with 4 sin 4 ; we shall find that cos 2 ^(2 + cos 2 /3)-4sin 4 ^ = 3-4sin 2 / 8~3sm 4 ^, and this is positive since sin 9 j3 is less than = . MAXIMUM SOLID OF REVOLUTION. 87 Next consider the numerator. (1 + a?)' (2a? - 1) ~ where x is put for Je 2 (1 4-> 2 ) -p 2 . The integral certainly increases with e if ; - \ - - increases with x. the differential coefficient of this with respect to x is - 4 , x which is positive ; for at? = e 2 (1 e 2 ) p 2 and is therefore less than unity. (3 sin 2 ff - 1) cos 2 /3 _ (3 sin 2 ft - 1) (1 - sin 2 ) sin 3 /3 sin 3 /3 4 1 = 3 sin ft + - 5 r-jf-Q ; sm /:? sin p the differential coefficient of this with respect to /3 is cosfi (3 - 4 sin*/3 - 3 sin 4 g) sin 4 /3 now this is positive when /3 = 0, and does not change sign so long as sin 2 /9 is less than - , in fact not until sin 2 f$ is greater than -= . O A Thus denoting the numerator of a? (e) by T we have shewn 7/T7 that.^,- is positive. dV 94. Thus we have shewn that -^ is always positive. If this had not been established we should have been in the following position ; instead of knowing that V continually increases from to x , we could only assert that V changes from to x . -Thus 88 MAXIMUM SOLID OF REVOLUTION. the same value of Fmight result from different values of a ; and in consequence corresponding to a given volume there might be more than one minimum surface. In general we should expect that these surfaces would differ in area, so that although both or all were minima surfaces every one could not be the least. This result would not be in any way repugnant to the principles of the Cal- dV culus of Variations. However by our demonstration that da always positive we see that our solution is unique. 95. As we have already stated the extreme case of our solu- tion is that in which we have for the generating curve a semicircle on the given part of the axis as diameter; see Arts. 81 and 89. When the given volume is greater than corresponds to this case the required solution is that given in Todhunters History of the Calculus of Variations, page 410. We shall however now add some remarks similar to those in Arts. 89... 94, in order to shew that the conditions relating to the constants can be satisfied. 9G. The boundary which we are now about to consider is composed of two equal straight lines AE and BF, and the curve EDF. The straight lines are at right angles to the axis at the given points, and they touch the curve at the points E and F. The curve is determined by the equation V (!+/>*) where a and c are constants. (1), MAXIMUM SOLID OF REVOLUTION. 89 The curve is traced out by the focus of an hyperbola as the hyperbola rolls on the axis of x : see the memoir by Lindelof, already cited in Art. 87. The curve consists of an endless repe- tition of portions like that in the diagram. HGr is the tangent at H, and is equal and parallel to AE. And c 2 = a 2 (e 2 1), where e is the eccentricity of the hy- perbola. Moreover the following relations hold : r sin- 1 - + 2a V (1 - e* sin 2 0) dB, J o r sin- -2a Jo 97. We take AB to be fixed, and we shall shew that a and e increase together; and that as they increase we obtain a set of boundaries like AEDFB the curved part of each of which is out- side that of its predecessor. Let AB 2h. When c = the boundary consists of the semi- circle From the general formulae of Art. 96 we have arid as AB=2h we have 2h less than 4a, and therefore h less than 2a. The curve (1) begins by being above the curve (2) at the points in the neighbourhood of A and B. Suppose if possible that (1) and (2) could intersect. At the points of intersection nearest to the axis of x let^ be the value of p for the curve (1), and p z the value of p for the curve (2), that is for the semicircle. Then p 2 is greater than p l ; and as the curves intersect we have at the com- mon point L fn "frjv-/ ^7- - J ^i i t -:*^ 90 MAXIMUM SOLID OF REVOLUTION*. But this is impossible ; for by what has been shewn -r-. - Y is less i than Thus any curve given by (1) with the relations of Art. 96 is outside the curve given by (2). 98. We have thus compared the curve (1) with the curve (2) ; we now proceed to compare together two curves determined by (1) with different values of the constants a and e. Since i h = a + a I e V(l & sin 2 0) dd, i . 2 0"> 5. tv 1 fi sin = - oa we have this shews that a and Se have the same sign, so that a and e in- crease together. Therefore of course c also increases with a, for Now take the two curves suppose j greater than a 8 , and therefore c t greater than c 2 . Then the former curve is above the latter at the points in the neighbour- hood of E and F\ and so by the method of argument already used in Art. 97, the former curve is entirely above the latter for the portion with which we are concerned. Thus we see that as a increases the volume generated by the revolution of our boundary continually increases ; and so a boundary exists for any assigned volume which is greater than that of a sphere having A B as diameter. MAXIMUM SOLID OF REVOLUTION. 91 99. We may establish in another way the result obtained in Arts. 91... 93, namely that when the volume is less than that of the limiting sphere the volume continually increases with a. Wehave where c 2 = a 2 (1 e 2 ). Now if c decreased as a increased we should see by the method of argument used in Art. 97 that the two curves ' could not intersect during the range of values with which we are concerned, that is the range between the points P and Q of the diagram of Art. 83. For suppose a l greater than 2 ; then e t is greater than # 2 ; 4 4 thus - (1 e *) is less than = (1 e 2 2 ), and so the angle of inclina- o o tion of AP to AB is greater for the curve corresponding to a l and e 1 than for the curve corresponding to a 2 and e z . Hence at the points in the neighbourhood of P and Q the former curve is above the latter curve. Then if the two carves could intersect, we should have at a common point, as in Art. 97, but, as PI is less than /> 2 at the lowest point of intersection, if c t is less than c 2 , the left-hand member of this supposed equation would be greater than the right-hand member. Thus the curves could not intersect. But c -y- = a (1 e 2 ) ea? -y- , so that -7- is negative when e = 1, da da da Therefore as we approach the limiting case of the semicircle for the generating curve, the curve for which a and e have specific values is necessarily inside the curve for which a and e have in- finitesimally greater values. 92 MAXIMUM SOLID OF REVOLUTION. If it be possible let a t and e 1 denote such simultaneous values of a and e that the corresponding curve is entirely inside the curve for which the simultaneous values are a x 4- Sc^ and e : + Se^ while it is intersected by the curve for which the simultaneous values are a t S^ and e t 8^. The last curve is below the curve cor- responding to j and 6j both at the vertex and at the points in the neighbourhood of P and Q, where the curve touches the straight lines : hence the supposed intersections must be at points intermediate between P and Q and the highest points. But this is impossible, for when Sa t and ^e t are small enough the curve corresponding to a x and e l would intersect the curve corresponding to j + Sa : and e l 4- ^ at the same point : but by supposition the last two curves do not intersect. 100. It is obvious that the whole problem illustrates our principle that discontinuity arises from a condition imposed. In Arts. 73... 79 there is a condition implicitly imposed, namely that the generating curve is to be entirely on the positive side of the axis. In Arts. 80... 99 we have the condition of concavity ex- plicitly introduced. 101. I proceed now as stated in Art. 86 to consider whether S is really a minimum. If we ascribe a variation to the rectilinear parts of the boun- dary we are certain that SS is a positive quantity of the first order, except in one particular case in which BS to the first order is zero. We will now apply Jacobi's method. MAXIMUM SOLID OF REVOLUTION. 93 /* / SM 102. Let u = l\ 2y VO- + P*) - - [ dx, the integral being taken J { a ) between fixed limits ; and suppose that we transform by Jacobi's method the term of the second order in Su. In order that there may be a maximum or minimum value of u we must have (1), then x^ + d .............................. (2). Here c t and c 2 are arbitrary constants. Suppose p found from (1) and substituted in (2) ; then (2) becomes the relation between x, y, Cj and c 2 , which is equivalent to the equation y=/ (a?, c v C 2 ) of Art. 25. Hence to find f and dc l j- , we have from (2) f 1 dp , 1 df -' + ' P c * Thus ^.- dc l p c v df -f = p. dc^ From (1) we have sothat 94 MAXIMUM SOLID OF REVOLUTION. Hence the value of z required in Art. 25 is theoretically found. Although we cannot express ~- explicitly in finite terms yet the form given for it will suffice for our purpose. 103. Now if we extend the process of Art. 84 so as to include terms of the second order, we shall have BS = 27r [I J Jw( l +P where the former integral extends over the rectilinear parts of the solution, and the latter integral over the whole solution. We are concerned now only with the latter integral. By Art. 23 this can be transformed to Now, in the manner of Art. 24, we have where m stands for -^ and v for -** ; and we are sure of a 4 minimum provided that v does not range over all values between positive infinity and negative infinity. Here . f f 2aJ in this expression put for y its value in terms of p as in Art. 90; thus we obtain MAXIMUM SOLID OF REVOLUTION. 95 If p is very small we have that is fl Thus if p is indefinitely small and positive, v is negative and numerically indefinitely large. As -=- is positive v increases with p ; thus two cases exist : either v becomes positive before p arrives at the greatest admissi- ble value, which with the notation of Art. 90 is tan /9, or v remains negative up to the limit. Moreover v changes sign with p. Hence in the second case v does not range over every value between posi- tive infinity and negative infinity ; and so we are certain of a minimum. In the former case however v does range over every value between positive infinity and negative infinity; and so it might appear that there is really not a minimum. The inference however would be erroneous ; all we can say is that Jacobi's trans- formation becomes inapplicable. Jacobi's method is quite satis- factory for problems of absolute maxima and minima values, but not for problems of relative maxima and minima values. In the present problem we have the important condition that the volume is to be constant; this imposes certain restrictions on &y: for instance By cannot be positive throughout or negative throughout. 104. Let us take a special case. Suppose e = 1 : then ~ Put tan for p ; thus dB cos sin 2 6 ' It will be found that If 1 1 , 1 + sin 0) .96 MAXIMUM SOLID OF REVOLUTION 1 . *7T* In this case ft~'> an( ^ so v ranges from positive infinity to negative infinity while 6 ranges between and /3, and again while ranges between and ft. Thus if Jacobi's method were ap- plicable we should infer that there is not a minimum ; but in this case our solution becomes a sphere, which we know gives the least surface with a given volume. 105. The general value of v becomes by putting tan 6 for p JL [ dd ~2aJ sin 2 0V(e 2 -sin 2 0) ; 1 f de 1 f dO f/ n sin 2 0y 4 ,| v= ^ej^re + ^-e]^ei( 1 --^-) ~ l \> therefore the sign of v is the same as that of The last integral is always finite ; we may suppose it taken between the limits and 0, so that it vanishes with 6, and increases with 0. Thus if be positive we see that v is oc when 6 = ; at /4 _ 4g 2 the limit ft we have cot = cot /& = * / 7-5 r * an d so the sign of v may be positive or negative at this limit according to the value of e. 106. Suppose the value of e to be such that v = when p = tan ft ; then v just ranges between positive infinity and nega- tive infinity. Thus if Jacobi's method were applicable we should infer that there is not a minimum ; we shall shew however that there is certainly a minimum. Whatever be the system of values of by we must have Sy vanishing at some point or points besides the extreme points ; otherwise Sy would be of the same sign throughout, and this is im- possible. Suppose that Sy vanishes for the point of the curve at which p CT ; take m such that 1 + mv vanishes at this point. Thus z vanishes when p ta-> and does not vanish at any other MAXIMUM SOLID OF REVOLUTION. 97 point. Hence this value of z satisfies the conditions involved in the investigation of Jacobi's theorem ; and thus the term of the second order in SS takes the essentially positive form given in Art. 103. 107. We may observe that our solution is certainly a mini- mum when compared with all such solids as have the same vertex as itself. For with this condition we always have Sy zero when p is zero. Hence instead of taking the general value of z we may take the particular value C 3 p, where C 3 is any constant, which will / suit Jacobi's method. Then = ^ , and the term of the second z p order in $S becomes and this is essentially positive. 108. The conclusion of the whole investigation is that we are sure of a minimum in some cases, and no argument can be drawn from Jacobi's method to shew that we fail to secure a minimum in any case. 109. It is easy to verify that zp satisfies the differential equation of Art. 23. Put p for z ; thus we get Pp -y- (Qq), that is we must shew that this expression vanishes, %ay Now - - r = y + c. a+/)* therefore ^ topqy therefore hence by differentiating we have the required result. 98 MAXIMUM SOLID OF REVOLUTION. We might attempt to get the complete value of z from the differential equation, as we thus know a particular value. The differential equation is, by Art. 23, d D dQ dz n d 2 z that is, P,____(2_ = o. Assume zpv\ then since we know that Here Q stands for Thus ^+^ + ^ = 0; dx therefore -j- Qp* = constant = 2 (7 say ; therefore , = 2(7 = py This will be seen to be just equivalent to what we found before in Art. 102. [It may be expedient to draw attention to the precise enun- ciation of the problem which has been discussed in Arts. 79... 94 and 101... 108. We have to determine the solid of revolution of minimum surface and given volume, under certain conditions : see Arts. 80 and 81. The condition, that the generating curve is never to be convex to the axis of revolution, is practically MAXIMUM SOLID OF REVOLUTION. 99 equivalent to the condition, that the generating curve is always to be concave to the axis of revolution : it is this condition which constitutes the difficulty and the interest of the discussion. At the period of the controversy to which I have referred in Art. 73, the Rev. Joseph Horner of Clare College, to whom I have been frequently indebted for valuable communications on other branches of mathematics, suggested to me to undertake the in- vestigation of the problem with this condition of concavity. I convinced myself then that the solution must consist of some combination of straight lines with a curve ; but I did not obtain the definite result until I returned to the subject for the purpose of the present essay. Some further remarks on the problem, which complete the application of Jacobi's method to it, will be found in Art. 288.] 72 CHAPTER VI. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 110. I WILL now take another problem which presents points of interest similar to those of the preceding problem, but which involves less complicated analysis. 111. The volume of a solid of revolution which cuts the axis at two fixed points A and B is given : determine the form of the solid so that the moment of inertia round an axis at right angles to AB through C the middle point of AB may be a minimum. Take C for the origin, and CB for the axis of x. Then the moment of inertia of the solid round the axis of y is and the volume is TT \y*dx\ the limits of integration being the values of x at A and B respectively. Hence by the usual theory we have to find the minimum of the limiting values of the variables being fixed, and a being a con- stant at present undetermined. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 101 Denote the integral by u ; then to the first order (?/ 2 + 2x 2 - 2a 2 ) fy dx. Thus we obtain as the necessary solution The interpretation is similar to that given in Art. 73. The solution consists of an oblate spheroid found by the equation which is connected by a straight line, namely, part of the axis of revolution, with the fixed points. The constant a must be 'deter- mined so that the volume of the spheroid may have the given value. For facility of conception we may consider the connecting straight line as an infmitesirnally slender cylinder. There is however this difference : in the present case on ac- count of the factor y the value of Bu to the first order is always zero, for the discontinuous solution, instead of being a positive quantity of the first order : see Art. 74. 112. The term of the second order in the value of &u is For the part of the solution which consists of the oblate spheroid this reduces to because here ^- + x z a? 0. For the other part of the solution the term of the second order reduces to because here y = ; and in this case a? 2 is greater than a 2 ; hence the aggregate term of the second order in Bu is necessarily positive ; so that we have a minimum as required. 102 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. And as tliere must be a least value, and no other minimum value can be found than that here given, we may be sure that we thus obtain the least value. 113. If the given volume be AC* the solution consists of the oblate spheroid alone, and there is no discontinuity. But if the given volume have not this value there will be discontinuity. STT If the given volume be less than -~- AC 3 the solution re- sembles that of the lower diagram of Art. 76. If the given volume be greater than AC 3 the solution resembles that of the upper diagram of Art. 76 ; unless indeed there is the condition that the solid may not stretch beyond the straight lines drawn at right angles to the axis drawn through A and B respectively. This case we will now consider. 114. The following will be the solution in this case : AD and BE are straight lines through the fixed points at right y angles to the axis. DFE is an arc of an ellipse given by y* + 2a 2 = 2a 2 . The required generating curve consists of AD, DFE, and EB. The constant a must be determined so that the volume generated by the revolution of this figure may have the assigned value. To justify this solution >11 that is necessary is to observe that the term of the first order in the variation of the moment of inertia is zero, and the term of the second order is positive, PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 103 115. In the problem discussed in Arts. 73... 78 we were able to confirm the proposed solution by appealing to the admitted fact that a sphere is the figure of greatest volume within a given surface. In the present problem we can confirm the proposed solution in an equally decisive manner. t Suppose an element of area indefinitely small in every direc- tion situated at the point (#, y] ; let this generate a ring by revolv- ing round the axis of x. Let m denote, the mass of this ring ; then the moment of inertia of the ring round the proposed axis is f \ + 3? } - Hence the solid of least moment of inertia with a given volume cannot be generated by any curve except the curve ~ + x" constant. For if there were any other generating curve we could obtain a less moment of inertia by removing matter which might be outside a bounding surface corresponding to this equation, and putting such matter inside the surface. The method here indicated has been already applied to problems relating to solids of greatest attraction : see Todhunter's History of the Cal- culus of Variations, Arts. 322 and 423. 116. It may be objected to the solutions of the present Chapter that our boundary has abrupt changes of direction, and a question may be raised as to the possibility of finding a solution which does not involve this abrupt change. I say that it is hope- less to seek for such a solution ; the reasons for this assertion have been already stated : see Art. 14, Remark in. 117. Let us proceed now to consider the problem with con- ditions explicit!} 7 imposed like those in Art. 80. We suppose that the given volume is less than A C 3 ; and we require the solid o of revolution to have this volume and a minimum moment of inertia, the generating line being never convex to the axis of revolution, and having no abrupt changes of direction. As in Art. 79 I venture to request any reader of these researches to investigate the problem for himself before examining the solution which I shall now propose. 104 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 118. As in Art. 82 every kind of boundary of the generating figure is excluded except straight lines and the ellipse y z + 2x 2 = constant. On trying various combinations I arrive at the conviction that there is no solution which exactly corresponds to the enunciation; but there is a limiting form to which we can approach as closely as we please : that is, we can find solids fulfilling all the required conditions, and having their moments of inertia never less than, though only infinitesimally greater than, a certain definite value. 119. Let ^IPand BQ be equal straight lines, making equal angles with AB. Let PD Q be an arc of an ellipse y* + 2a? 2 = 2a 2 . y Let the constant a, and the angle PAC be determined so as to make the volume generated by the revolution of APD QB round the axis of x equal to the assigned volume, and also to make the moment of inertia of the solid round the axis CD the least pos- sible : this of course is only an ordinary problem in the Differential Calculus which we may assume to be capable of solution. Then I say that the boundary thus determined is the limiting form of the solution to our problem in the Calculus of Variations. It does not strictly fulfil the conditions because there are abrupt changes of direction at P and Q ; but we may suppose a curve drawn indefinitely near to this boundary so as to avoid the abrupt change of direction, and to be always concave to the axis. See the diagram to Art. 14. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 105 120. We proceed to justify the statement of the preceding Article. We have to the first order $u = fy (/ + 2j; 2 - 2a 2 ) fydx, and as y* + 2# 2 - 2a 2 = for the arc PDQ this reduces to where the integral is to be taken from the value of x correspond- ing to A to the value of x corresponding to P, and then from the value of x corresponding to Q to the value of x corresponding to B. Now if fy be proportional to y this expression for $u will vanish ; for by supposition AP and BQ are so taken as to make the moment of inertia a minimum corresponding to the assigned volume, and therefore of course an infinitesimal change in the position of AP and BQ must leave the value of u unchanged to the first order. In fact instead of saying that a and the angle PA C are to be determined so as to make the moment of inertia a minimum for the assigned volume, we might say that they are to be determined so as to produce the assigned volume and to make vanish between limits corresponding to AP and QB when fy is proportional to y ; that is, y 2 (y* + 2x* - 2a 2 ) dx must vanish between the limits corresponding to AP and QB. 121. We remark in passing that since the last integral is to vanish the expression y 3 + 2x 2 %a a must change its sign within the range of integration ; this shews that if the arc PDQ were continued it would cut the straight lines AP and BQ. Hence it follows that the tangent at P to the arc PD Q is inclined to the axis of revolution at a less angle than A Pis. This is essential to our solution, in order that when we draw a curve close to the boundary APDQB, as explained in Art. 119, this curve may always be concave to the axis. 106 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 122. The demonstration that we have a minimum is similar to that in Art. 85. The straight line AP is supposed to be the same as in Art. 119 ; and ALK is a curve obtained from AP by ascribing admissible values to By ; so that the curve has no convexity towards the axis of revolution, and no abrupt change of direction. 2 The point Fis not found by making AF= -- AP as in Art. 85 ; o but F is the point where the straight line AP is cut by the curve 2 + 2x 2 = 2a 2 . Thus in passing from AP to the curve ALHK we have ulti- mately a gain of positive elements corresponding to the area ALN, a relief from the negative elements corresponding to the area HIPG, and a gain of positive elements corresponding to the area IKP. In like manner the other two diagrams of Art. 85 apply here. 123. Hence our conclusion is like that in Art. 86. We find that Bu is always a positive quantity of the first order, and so we are sure of a minimum ; except in one particular case, namely, that in which for the rectilinear part of the boundary By is taken proportional to y, and then Bu is zero to the first order. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 107 In this particular case then we must examine the term of the second order in Sw; as we have seen in Art. 112, this term is Kow for the part PD Q of the boundary this reduces to which is positive. For the parts AP and B Q we have by suppo- sition y proportional to y, so the sign of the term is the same as that of and therefore is the same as that of for we know that for the limits with which we are concerned Hence the term of the second order is positive. Thus we can assert confidently that the proposed solution is a minimum. And as there must be a least value, and no other minimum value presents itself besides this, we may conclude that this is also the least value. 124. In the discussion of the present Chapter we have the discontinuity of a straight line and a curve which meet without touching. As in Art. 100 we see that the discontinuity arises from a condition implicitly imposed, or explicitly imposed. 125. We may give a still more simple example of the kind discussed in the present Chapter. Find a curve which shall cut the axis of x at given points, and enclose an assigned area, and have a minimum moment of inertia round an axis at right angles to the plane of the curve passing through the point on the axis of x which is midway between the two given points. 108 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. Take the axes as in Art. Ill ; then proceeding in the usual way we find that we require the minimum of - 2 ydx. Denote this by u ; then to the first order 2 ~a 2 ) Sydx. Thus we obtain as the necessary solution tf + cc* - a 2 = 0. This is the equation to a circle having its centre at the origin. We obtain the same kind of discontinuity as we had in Arts. 73 and 111. The present case resembles that of Art. 73 in the circumstance that $u instead of being zero to the first order for the discontinuous solution is a positive quantity. The investigations of Arts. 113... 123 apply with obvious modi- fications to the present problem. The simplicity of the present problem recommends it as offering an easy case for the considera- tion of any person who might wish to contest the conclusions at which we have arrived. 126. Some variety might be introduced into the problems of Arts. Ill and 125 by giving a different position to the axis about which the moment of inertia is required. Instead of cutting the plane of x, y at the origin let this axis cut the plane at the point (h, &). Then in Art. Ill we have u = (a- A) a + fc"- aj tfdx; and in Art. 125 we have For in the latter case we may consider separately the parts of the figure which are above and below the axis of #; as it is obvious that each part must have the minimum property. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 109 The centre of the ellipse in Art. Ill, and the centre of the circle in Art. 125 is now not at the origin but at the point (h, 0). It is easy to see what effect this will produce on our solution. 127. Suppose another condition imposed besides those in Art. 117 : let the generating curve be required to have no abrupt change of direction, to be never convex to the axes, and to cut the axis at given points and at given angles. For simplicity suppose these angles equal and denote them by 7. Let the problem as enunciated in Art. 117 be solved; and suppose that the boundary there obtained cuts the axis at an angle /3. I. If /3 = 7 the solution of the problem in Art. 117 obviously satisfies all the conditions of the problem of the present Article. II. If /3 is less than 7 this solution must still be made to suffice. In the diagram of Art. 119 we must conceive straight lines making an angle 7 with AB to be drawn, one through a point to the right of A but indefinitely close to A, and the other through a point to the left of B but indefinitely close to B. Thus we obtain a boundary illustrated by the diagram ; AE and BF will be indefinitely short. As in Arts. 118 and 119 the I problem most strictly speaking does not admit of a solution. But the diagram gives the limit towards which we must approach as we make the moment of inertia continually smaller while we retain the conditions imposed at the commencement of this Article. 110 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. III. If {$ is greater than 7 the solution of the problem in Art. 117 will be of no use in the present case. Our solution will then be as follows : Through A and B draw straight lines inclined at an angle 7 to AB. Let AP and BQ denote these straight lines. Let PD Q be an arc of the ellipse y* + 2# 2 = 2a 2 ; and let a be so taken that the volume generated by the revo- lution of APD QB round AB may have the assigned value. To shew that this is a solution all that is necessary is to examine the value of Su. Let P'D' Q' be a curve obtained by variation from PD Q, and suppose that P' and Q' are below P and Q respectively. To the first order - 2a 2 ) Sydx. Now for limits corresponding to P and Q this vanishes. For the small portion between P and P', and that between Q and Q', we have % negative and y* -f 2a; 2 2a 2 negative ; so that there is a small positive element of Su: this would vanish if AP touched PQ at P. The term of the second order in Su is positive as in Art. 111. If the curve obtained by variation has its ends above P and Q respectively, then &u to the first order vanishes entirely, and the term of the second order is positive as before. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. Ill Thus we have a minimum. Of course the given volume must be less than that of the double cone, which would be obtained by revolving round AB the triangle formed by producing AP and BQ to meet. 128. The discussion in the preceding Article suggests to us to try the effect of imposing another condition on the problem of the preceding Chapter ; thus we have the following enunciation : determine the solid of revolution of minimum surface, and of given volume, supposing that the generating line is to cut the axis at given points, at given angles, and is never to be convex to the axis, and to have no abrupt change of direction. 129. For simplicity suppose the given angles to be equal, and denote them by 7. Let the problem as enunciated in Art. 83 be solved; and suppose that the boundary thus obtained cuts the axis at an angle ft. I. If ft = 7 the solution of the problem enunciated in Art. 83 obviously satisfies all the conditions of the problem of the present Article. II. If ft is less than 7 the solution must still be made to suffice : the mode in which this must be effected is the same as that in Case n. of Art. 127. III. If ft is greater than 7 the solution of the problem in Art. 83 will be of no use. I propose the following as the solution : Take a curve determined by . a ^ ~ = y* + c, where a and c are constants; and draw tangents at the points where the tangents will be inclined at an angle 7 to the axis of x : then there are two conditions which must serve to find the values of the con- stants ; namely, the tangents must intersect the axis of x at the given points, and the solid formed by the revolution of the boun- dary round the axis of x must have the assigned value. It remains then to shew that these conditions can be satisfied, as well as the condition necessary for a minimum. 112 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 130. Let S denote the surface of the solid generated in the manner described ; then we must shew that BS to the first order is zero or positive : this can be done in the manner already ex- emplified. As in Art. 84 we can shew that where the integral is to be taken between limits corresponding to the rectilinear parts of the boundary : for these parts we have = COS 7' Now the condition that there is to be no convexity in fact renders $y necessarily negative as far as we are concerned with it. If the ordinate at the points common to the rectilinear and curvilinear parts is -^-0057, we may by taking %y proportional 2i to y as an extreme admissible case just make BS zero to the first order. If the ordinate at the specified points is greater than cos 7, then a gain of positive elements is secured, and $S is positive to the first order. Then if By be taken in any other admissible way, $S is positive to the first order : see the reasoning and the diagram of the last of the three cases of Art. 86. Thus BS is zero or positive to the first order provided the la 2 ordinate at the point of discontinuity is not less than cos 7. If the ordinate is greater than cos 7, we are sure of a mini- mum even without looking at the terms of the second order. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 113 131. Now let us consider the conditions which the constants must satisfy. Put the equation to the curve in the form so that a and e are the constants. Let y l be the ordinate of the point which is common to the rectilinear and the curvilinear part of the boundary. Then, by Art. 90, we have y^ Gb cos 7 + a V( g2 ~ sm * T) .................. (1) ; and this is not to be less than -=- cos 7 ; 2t therefore V (e 2 sin 2 7) not less than -= cos 7 ; therefore e 2 not less than sin 2 7 4- 7 cosfy Let 2h denote the distance between the given points ; then /tany Ldp ........................ (2), j where L has the meaning assigned in Art. 90. Let 2F denote the volume generated by the revolution of the boundary round the axis of x ; then COt 7 sT7-,7 /0\ f 3 Hdp (3), where H has the meaning assigned in Art. 91. Suppose the value of y v from (1) substituted in (2) and (3) ; then from (2) we find T = a function of e and known quantities. Sub- stitute this value of a in (3) ; thus F becomes a function of e and of known quantities. And as e changes continually from an indefi- nitely large value we see that F will vary continually, as long as e is greater than sin 7. 8 114 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. At the extreme case of e infinite we have a indefinitely small ; and ae cot 7 = h. Then V * - where y cot 7 = h. o The assigned volume must of course not exceed - - or o the problem will be impossible. = A/ sin 2 7 + ^ At the value e = A sin 2 7 + ^ cos 2 7 the value of V will be the same as we should obtain in Art. 91 if ft were changed into 7. Now this value is less than the assigned value of the present pro- blem ; for this assigned value corresponds to that obtained in Art. 91 with the value ft which is greater than 7 : and we know that the value of V in Art. 91 increases with e, and is therefore greater for ft than for 7. Hence finally as from (3) the value of F ranges between limits which include between them the assigned value of the present problem, it will be possible to take e such that F shall just be equal to the assigned value. 132. We may feel sufficiently secure that there is only one solution thus : The greatest possible assigned volume corresponds dV to e infinite and a indefinitely small ; then if -=- be the differen- tial coefficient of F with respect to e when a is supposed elimi- dV nated we are sure that j- is positive when e is infinite. There- fore if e be diminished F diminishes. Suppose that -j- could vanish at any point ; then we should have the same value of F corresponding to two adjacent values of e. Thus we get two ad- jacent solutions. But this is impossible ; for by Art. 130 each is a true minimum, therefore each is less than the other. 133. We may remark a difference between this problem and that in Art. 117. Here the rectilinear and curvilinear parts touch, while there they met at a finite angle. The difference depends PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 115 on the circumstance that in the present problem the expression under the integral sign in the quantity to be a minimum involves the differential coefficient p, which did not occur in the former problem. 134. Another modification may be given to the problem of the preceding Chapter. A bowl is to be made in the form of a figure of revolution so as to have a given surface and to hold a maximum volume. If the problem be stated thus without any condition the bowl will have to be a sphere ; this must be considered a limiting case, as strictly speaking a closed surface cannot be called a bowl. But suppose the breadth of the bowl at the open end to be given. Let Ox be the axis of revolution; let 2k denote the given breadth, and suppose OB - k. The solution, as in Art. 73, is a circle which has its centre on the axis of x. Let the radius of the circle be r t and let OA, the depth of the bowl, = h. Then to find h and r we have the equa- tions = the given surface = 8 say ; S therefore h z = -- Jc\ 7T This gives always a real value for h, as $ must of course be not less than Trk*. 82 116 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. Next suppose that the depth of the bowl is also given. If this given depth is greater than that which our solution assigns, we must take this solution and suppose an indefinitely V narrow conical or cylindrical part attached at the bottom of the bowl, and having the same axis as the bowl. If the given depth is less than that which our solution assigns we may understand the condition in two ways. If we merely mean that the depth as measured along the axis is to have the given value, we use such a solution as arises from having the point of the conical part in the diagram inwards instead of outwards. But if we mean that the depth is at no point to be greater than the given quantity, then the solution must be composed partly of a straight line through A at right angles to the axis of x, and partly of a curve passing through B and touching this straight line, and satisfying the differential equation see Art. 73. 135. A curve generates a bowl by revolving through 180 round an axis which it cuts at two fixed points : find the curve so that the centre of gravity of the surface may be at the greatest distance from the axis, the area of the surface being given. The enunciation does not immediately suggest any difficulty or strangeness as likely to occur in the solution. PROBLEMS. ANALOGOUS TO THAT IN CHAPTER V. We have by the usual theory to find a maximum of 117 Call it u ; then we obtain ay - constant; the constant must be zero since the curve is to cross the axis. We cannot take y = a, for then we shall not have Sw = to the first order, since We must then attempt to form a solution by combining p = &> and y = ; the former gives a straight line parallel to the axis of y, and the latter is the axis of x. This indicates that strictly speaking there is no solution ; but there is a kind of limit towards which we may approach indefinitely. The generating curve must be supposed to run indefinitely close to the axis, except where it turns off, nearly at right angles to the axis, and returns again; thus ACDEB may represent it, CD and DE being very close to each other, and nearly straight lines. 118 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. The limit to which we approach consists of the straight line AB and a straight line FD at right angles to AB\ the position of F is arbitrary ; the length of FD must be such that may be equal to half the given surface. To shew that the limit is a true maximum, we may proceed as in Art. 64. We may observe that instead of a single straight line FD at right angles to AB we might take two or more. Suppose we take two, say FD and HK. Then we must have equal to half the given surface. This would still be a maximum. Instead of one long slender part CDE, as in the former figure, we should now have two. It is obvious however that the distance of the centre of gravity from the axis is greater when there is only one long slender part than when there are two or more such parts. For in passing from the former case to the latter, we in fact remove matter from one position and put it into another which is nearer to the axis ; and thus we bring the centre of gravity nearer to the axis. We may of course obtain the same result by using the common formula for the position of the centre of gravity. 136. Suppose we ask for the solution of this problem with the condition that the boundary is never to be convex to the axis and to have no abrupt changes of direction. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 119 Now, as in Art. 82, every kind of boundary of the generating figure is excluded by the Calculus of Variations, except straight lines, and curves which satisfy the differential equation if ay .. = constant. Proceeding as in Art. 83, I suggest the following for the solu- tion of the problem. Let A and B be the fixed points on the axis ; let AP&ud BQ D JL J3 be equal straight lines equally inclined to the axis which touch the curve PDQ at Pand Q respectively; and let PDQ be an arc of the curve determined by the above differential equation. The constants must be so taken as to ensure the tangency at P and Q, and to make the surface generated by APD QB have the given value. Moreover there is a condition to be satisfied which we shall investigate presently ; this condition connects the constant a with the ordinate of P and Q. After the full investigation in Chapter v. it will not be neces- sary to discuss the present problem in detail. We see that the expression for Su given in Art. 135 vanishes for the part PDQ, and for the parts AP and QB it reduces to a ,x , oyax. Let y l denote the ordinate of P and Q ; then as in Art. 85 we see that must be such that 120 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. ., thus Let $ denote the angle PAB or QBA ; then if we put c 2 for the constant in the above differential equation we see that At the highest point D we shall have so that it will be found on investigation that we must take the upper sign. 137. Given the mass of a solid of revolution of uniform density, required its form so that the attraction on a point in the axis may be a maximum. This is a well-known problem. Taking the origin at the point, and the axis of revolution for the axis of x we require the maximum of where a is a constant. Denote the integral by u; then to the second order PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 121 Thus the solution is to be found from x it will be convenient to write - - 8 for 2a ; thus (x* + y*}* = c*x. c When y = we have x = or c ; thus c is the length of the . The volume = TT I y*dx IT I {(c 2 a?)* a? 2 } efa? J ^ axs. 3 IN 47TC 8 so that the volume being given c is known. In the term of the second order in Su the coefficient of becomes ^ -a+^^.^t^-'Ci, 3 f ^'b/^ * so that the term is ^ I (^ ~~ c3 ) (^) 2 dx. This is negative as it should be. It is well known from other considerations that we have really a maximum. [See Todhunter's History of the Calculus of Varia- tions, Art. 322.] 138. Now let us impose the condition that the length of the axis shall have a given value. The solution already obtained will not apply ; because in this solution the length of the axis is determined as soon as the volume is given. We observe however that the term of the first order in Su vanishes if y ; and this suggests a solution of the kind already exemplified ; namely, the generating curve must be made up of (a? 2 + y*)% = c*x, together with a portion of the axis of x. 122 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. Let h denote the given length of the axis ; then the term of the second order in Su becomes The first term is negative, and so is the second whether c be less or greater than h. Thus we have a solution whether c be less or greater than h ; the case in which c is greater than h resem- bling that which has already presented itself in Art. 134 with the point of the cone turned inwards. But if c be greater than h the problem may be understood in another sense, and perhaps more naturally : namely, that the solid is not to extend beyond the ordi- nate at the point x = h. The solution then is given by where the constant 7 is to be determined from the condition rh TT I y*dx = the given volume. All the circumstances of the pro- ' blem are thus satisfied : it will be observed that there is no term outside the integral sign in the value of &u. The generating curve may be said to consist of the curve (x z + y z y* = y*x from x = to x = h, and of the ordinate at the point x = h. If any objection is taken to the abrupt change of direction at the point where the curve joins the straight line we must answer the objection in the same manner as before : see Art. 14 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 123 139. Suppose we attempt to introduce also the condition that the radius of the base shall have a given value &, the length of the axis being h as before. We shall have in some cases to employ the solution already given as a limiting form in the manner formerly exemplified. Thus suppose CA = h, CB = Jc ; and let the given volume be such that the curve ADE would have been the solution if h and k had not been mentioned. Then we must conceive a curve drawn indefinitely close to A DECS, and the closer it is drawn the nearer it will approach to the form which gives the greatest attraction under the specified conditions. Or instead of the solution of Art. 137 we may have to take the solution of Art. 138 and make it in a similar manner applicable to the problem with the additional condition we have now im- posed. There will be two varieties, because the base obtained in Art. 138 may be greater or may be less than that which is now prescribed, and thus there will be a difference in the mode of adjustment to the prescribed base. It may happen that in these solutions some of the ordinates are greater than k. 124 PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. But if we understand our condition to mean that we are to have no ordinate greater than k we shall require in some cases another solution, namety, a portion of the curve (x z -\- combined with a portion of the straight line y = k. The condition for finding 7 will now be TT I ifdx + 7r& 2 (h%) = the given volume, *^o where f stands for AE, so that ({* + tf)% = 7 2 . Here Bu to the first order vanishes with respect to AD ; and - D J? with respect to DB reduces to Pi- \ + ^5] fydx, J* I T 2 J + V*) ' that is to Now if x lies between f and h the expression ^ . - 1 (^ 2 -fF)^ is positive ; for this expression would be zero if we put instead of k the ordinate y of the curve corresponding to the abscissa x, and y is greater than k, so that the expression as it stands must be positive. And along DB we have fy essentially negative if it is not zero. Hence the above variation is a negative quantity of the first order unless y vanishes at every point of DB. In the ktter case we proceed to the term of the second order in the variation which reduces to | ^ ~ C ) (#) 2 dx 9 which is Jo 2c~3 negative. Thus a maximum is secured. PROBLEMS ANALOGOUS TO THAT IN CHAPTER V. 125 [We here confine ourselves to the case in which the curve AD, when continued beyond Z>, does not cut the straight line y = k again between D and B. It is easy to see what modi- fications have to be made if the curve does cut this straight line again between D and B] 140. The examples discussed in the present Chapter furnish numerous illustrations of the principle that discontinuity arises from conditions imposed in the problems. In Art. 139 we have a very simple example of the general principle of Art. 17. CHAPTER VII. BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 141. IN the present Chapter we shall discuss some cases of discontinuity which arise by imposing various conditions on the problem of determining the shortest course of a particle under the action of gravity. 142. Required the curve of quickest descent from a fixed point A to a fixed point B, supposing the moving particle con- strained to remain on a fixed smooth inclined plane which con- tains A and B. The curve is known to be a cycloid with its base horizontal, having a cusp at A ; in fact we resolve the force of gravity into two components, one at right angles to the inclined plane, and the other along this plane : then the former component may be disregarded. Now suppose we require the curve of quickest descent from A to B, when there are two fixed smooth inclined planes, one passing through A and the other through B, and the moving particle is constrained to move first on one plane, and then on the other, assuming that no velocity is lost in passing from one to the other. The required curve consists of two arcs of cycloids with their bases horizontal ; the first arc has a cusp at A ; the second arc must have its cusp so situated in the second inclined plane that the velocity of the particle at the commencement of this arc is that which would be acquired in falling from the cusp: hence the cusp will in fact be in the horizontal plane which passes through A. BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 127 It is easily seen by examining the term in the variation which is outside the integral sign, that at the point where the two arcs of cycloids meet their tangents must be equally inclined to the line of intersection of the two planes. 143. A particle falls from a fixed point A to a fixed point B, passing through another point C: find the entire path when the time of motion is a minimum (1) supposing C to be a fixed point, (2) supposing C constrained to lie on a given curve. [This problem was proposed by the present writer in the Mathematical Tripos Examination of 1866.] We assume that no change of velocity occurs at the point 0. (1) From A to C the path must be a cycloid having its base horizontal and a cusp at A ; from C to B the path must be a cycloid having its base horizontal and a cusp in the horizontal plane through A. This statement is evident from the combination of two known results ; one relating to the brachistochrone when the initial velocity is zero, and the other relating to the brachisto- chrone when the initial velocity has a given value which is not zero. If C is not in the vertical plane which contains A and B, the two portions of the entire path are in different vertical planes. (2) From A to C the path must be a cycloid having a cusp at A ; from C to B the path must be a cycloid having a cusp in the horizontal plane through A : each cycloid has its base horizontal. This statement is evident for the same reason as before. Also the tangents to the two portions of the path which meet at C must make equal angles with the tangent to the given curve at C : this condition serves to determine the point C. To demonstrate this we proceed thus : Let us first suppose that A and B and the given curve are all in the same vertical plane. Let the axis of y be vertically downwards, and the axis of x horizontal. Let x and y denote the co-ordinates of G. Let ty' (x) denote the value of -jt for the given curve at the point G ; let p Q and p l denote similar things for the two portions of the path which meet at G. Then, as in Todhunter's Integral Calculus, Art. 361, the part of the 128 BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. variation outside the integral sign reduces to +1 > ') and in order that this may vanish we must have this involves the condition stated above. If A and B and the given curve are not in the same vertical plane, the same condition may still be shewn to hold : see Art. 367 of Todhunter's Integral Calculus. 144. Required the curve of quickest descent from a fixed point A to a fixed point B ; supposing that a screen or obstacle is inter- posed between A and B, having a given finite aperture through which the path must pass. It may happen that the aperture is so situated that the cycloid of quickest descent from A to B passes through it : and then of course the condition imposed does not affect the solution of the problem. Suppose however that the aperture is not so situated, then, I say, that the required path must pass through some point of the boundary of the aperture. For if the path did not pass through the boundary there would be no limitation imposed on the dependent variable, and thus by the ordinary theory we are sure that we have not a curve of minimum time. Thus the path must pass through the boundary of the orifice ; and therefore the present problem is reduced to that of Art. 143. 145. Again : find the curve of quickest descent from a fixed point A to a fixed point B with the condition that the path must pass through some point C which is on a given fixed surface. It may happen that the given fixed surface is so situated that the cycloid of quickest descent from A to B crosses it ; and then of course the condition imposed does not affect the solution of the problem. Suppose however that the given fixed surface is not so situated. The position of the point C must by Art. 143 be such that the two parts of the path make equal angles with the tangent BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 129 to any curve drawn on the surface through the point of contact : the two parts must therefore make equal angles with the normal to the surface at that point, and must lie in the same plane with that normal. To shew this however from the ordinary formulas we proceed thus : Let p stand for -~ , and iz for -=- ; these being formed on dx dx' the curve. Let the subscript 1 apply to the end of the first portion of the path, and the subscript 2 to the beginning of the second portion. Let (x, y, z) = be the equation to the surface ; put v 2 for 1 -fj9 2 + d6 d$\ 1 (dd> , dd> , d6\ ~{^ + p-^-^^--r) = - :j +P:7 + OT :J - v^\dx r dy dz/ } v z \dx r dy dz) z For if we substitute for -/ and -f in terms of -~ from the two dy dz dx equations just given, we shall find that each member numerically = v z -v l dx' Of course this equal inclination of the tangents to the normal to the surface, and their lying in the same plane with it, might have been anticipated. For through an infinitesimal portion of the path we may consider that the velocity does not sensibly vary : and so, by a well-known geometrical fact, the path should in the neighbourhood of the surface resemble that of a ray of light re- flected at the surface. Suppose we require the curve of quickest descent from a fixed point A to a fixed point B, with the condition that the path must not cross an obstacle in the shape of a fixed closed surface. In this case, supposing the obstacle so situated that the ordinary cycloid is inapplicable, the solution will consist of three parts. The first part and the third part will be arcs of cycloids ; the second part will be the brachistochrone for a particle constrained to move on the given surface, and we need not discuss this, for it is BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 131 a well-known problem. The parts of which the solution consists will have common tangents at the points where they meet : this is obvious from what has already been given. [In the original essay there was some confusion between the two problems of the present Article, the second problem being enunciated and the first solved : hence some slight changes in the Article were necessary, and have been made.] 146. The problems which we have hitherto considered in this Chapter illustrate the principle that discontinuity arises from con- ditions imposed. The discontinuity here is that the first and the last parts of the path are arcs of different cycloids, generally in different planes, and meeting at a finite angle : in the second problem of Art. 145 there is however an intermediate arc be- tween the two arcs of cycloids. It is obvious that in all the problems of this Chapter there must exist some solution which gives the least time of motion ; and the Calculus of Variations shews that no other solution can exist than that which we have taken in each case respectively. Hence we need not consider the terms of the second order in the variation ; in fact, however, in all the cases except the first of Art. 143 we should find that the sign of the term of the second order cannot be ascertained by any theory at present known. 147. Required the curve of quickest descent from a fixed point A to a fixed point B\ with the condition that the particle is never to descend below the horizontal straight line through B. By the principles of the Calculus of Variations combined with our condition every line is excluded except a cycloid having its base horizontal and its cusp at A, and the horizontal straight line through B. If then the horizontal distance between A and B does not exceed ^- times the vertical distance, the required line consists of the cycloid alone : our condition does not come into operation. 92 132 BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. If the horizontal distance between A and B exceeds ^ times 2i the vertical distance, the required curve consists of the cycloid A G and the horizontal straight line CB, which is the tangent to the cycloid at its vertex C. 148. The example in the preceding Article resembles others which have already been discussed: the boundary which deter- mines the area into which the path is not to pass may itself form through some of its extent a part of the required solution. If the boundary instead of a horizontal straight line through B is any straight line, the solution is substantially the same. If a straight line through A is the boundary which the path is not to pass, the required line consists of A C, a portion of this straight line, and CB an arc of a cycloid; the cycloid must have its base horizontal and its cusp in the horizontal straight line through A y and must touch the bounding straight line at C. BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 133 If the boundary is a straight line which does not pass through A or B, and the single cycloid from A to B is inapplicable because it would cross the boundary, the solution consists of three parts : an arc of a cycloid having its cusp at A and touching the boundary; a part of the boundary ; and an arc of a cycloid which touches the boundary, passes through B, and has its cusp in the horizontal straight line through A: each cycloid has its base horizontal. 149. Required the curve of quickest descent from a fixed point A to a fixed point B, with the condition that the path is not to pass outside the given circular arc AB which does not exceed a quadrant, B being the lowest point of the circle. It will be convenient to work out the solution from the beginning. Take any point on the horizontal straight line through A for origin. Let the axis of x be horizontal, and the axis of y vertically downwards. We have to find a minimum value of Call this u : then where Jy (i So long as % is susceptible of either sign we know that there 775 cannot be a minimum unless N ,- =0: and this leads to a dx cycloid having its base horizontal and its cusp on the axis of x : these conditions as to the position of the cycloid must be under- stood in all that follows with respect to the present problem, although for the sake of brevity we may omit to state them explicitly. Hence the curve we require can consist of nothing except a portion or portions of such a cycloid, together with a portion or portions of the circular arc. See Art. 18. BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. It is obvious that the solution cannot consist of a cycloid alone, as if there were no condition imposed : for such a cycloid would have a cusp at A t and therefore its tangent vertical there ; and so it would be initially outside the circle. 150. We therefore proceed to investigate whether the neces- sary conditions will be satisfied by the curve composed of a circular portion AH, a cycloidal portion US, and a circular portion SB ; or of two out of these three possible portions ; or of one only. Let be the centre of the circle ; draw A C horizontal : let OA = r, 00 = b. The equation to the circle is c being a constant depending on the point in A which we take as origin. For the circle we have AT ' 7~ *& ~~ @ ry and so it will be found that N _dP dx For the cycloid we know that tf-f-o. dx BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 135 Our object is to ensure that Bu shall be positive. The value of By corresponding to any part of the circular arc is necessarily nega- tive; so that |(^ -JT) fydx\& necessarily positive for limits cor- responding to the circular arc provided y b is negative. 151. Now we observe in the first place that the circle and the cycloid must touch at E, supposing that E does not coincide with A. For let P be the value of P corresponding to the circle at E, and let P l be the value of P corresponding to the cycloid at E. Then the part of &u which is free from the integral sign gives rise at the point E to the term (P P,) By. Now By is necessarily negative at E ; and therefore P P x must be zero or negative : it cannot be negative, for then the cycloid would fall outside the circle. Hence P P l must be zero ; and therefore the circle and the cycloid must touch at R. Of course if R coincide with A we have only P i By for the corresponding part of Bu, and this vanishes since By vanishes : thus it is not necessary that the circle and the cycloid should touch. In like manner if S does not coincide with B the circle and the cycloid must touch at 8. 152. We shall now consider what results follow from suppos- ing the circle and the cycloid to touch at R. See the diagram to Art. 150. Let OR and AC intersect at T\ let the angle OTG be de- noted by 6, and the angle OA C by a. Thus -nm b rsina RT=r--ra=r ^^ . sin 6 sm 6 Now from the properties of the cycloid we know that the diameter RT . r (sin d- sin a) of the generating circle is -, ^ , that is r-^ . sin 6 sin If the cycloid touches the circle again at 8 we obtain another expression for the diameter of the generating circle by ascribing to the value which it has at 8 instead of the value which it has at 136 BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. R : and of course these two expressions must be equal in value. This leads us to examine the values of which the expression sin 9 sin a . r-r-rt is susceptible. sm'0 Denote it by v ; then dv _ cos (2 sin a sin 0) 50 = ~~sii?T~ If 2 sin a is less than unity the maximum value of v is given by sin 6 = 2 sin a ; and then v may twice have an assigned value, namely, once when 6 is less than the value which corresponds to the maximum, and once when is greater than this value. If 2 sin a is not less than unity then the maximum value of 6 is given by cos = ; and as changes from a to = the value of v continually increases. In this case then we do not get the same value of v for two admissible values of 6 ; and it is impossible to draw an arc of a cycloid RS which touches the circle at two points. Nor is it possible to draw an arc of a cycloid which touches the circle at a point R and passes through B. For the diameter of the generating circle would be here less than the maximum value of r(sin0 sina) , . , . . . . . . . a . , that is less than r (1 sin a), that is less than UJf 9 sin (/ and so the cycloid could not pass through the point B. Hence we arrive at the result that the circular arc itself is the line of quickest descent. It will be observed that y b is equal to r (sin 6 2 sin a), and this is in the present case negative for the whole of the circular arc; and so \(N -j ) 8 y dx is necessarily positive for admissible values of By. Here there is no discon- tinuity. 153. Let us now return to the case in which 2 sin a is less than unity. In this case at B and adjacent to B we have y b positive for the circle, and so I ( N -y-J &y dx would be negative for admissible values of Sy. Hence the arc of the circle adjacent BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 137 to B cannot form a portion of the required line of quickest descent. Thus the arc of the cycloid must pass through B. Hence finally, if 2 sin a is less than unity the required curve consists of a portion AR of the circle, and then a cycloidal arc touching the circle at E and passing through J9; that is, we have the discontinuity of the arcs of two different curves which touch at the common point. And the common point R must be such that here y is less than b ; and in fact this is ensured by the nature of the cycloid. For as the cycloid is to touch the circle internally the radius of curvature of the cycloid must be less than the radius of the circle ; that is twice TR must be less than OR ; therefore TR is less than TO, which makes y less than b for the point R. It is obvious that 2 sin a is greater or less than unity according as the circular arc BA is less or greater than an arc of 60. 154. It may be useful to shew that it is possible to draw such a cycloid as we have supposed when 2 sin a is less than unity. Suppose we take r b for the diameter of the generating circle of the cycloid, and put the vertex of the cycloid at B ; this cycloid will fall without the circle at B, because 2r - 26, which is the radius of curvature of the cycloid at the vertex, is by supposition greater than r. On the other hand, if the diameter of the generating circle of the cycloid is indefinitely great, and the cycloid be made to pass through B, it will obviously fall entirely within the circle. Starting from the last case diminish the diameter of the generat- ing circle of the cycloid continuously, making the cycloid always pass through B. Then we must arrive at the case in which the cycloid just touches the circle before cutting it. The point of con- tact will not be at A, for the tangent to the cycloid would then be vertical, while the tangent to the circle would not be vertical. The point of contact will not be at B, for there the tangent to the circle is horizontal, while the tangent to the cycloid would not be horizontal. Hence the contact must take place at some interme- diate point, as we require. 155. Required the curve of quickest descent from a fixed point A to a fixed point B, with the condition that the path is not 138 BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. to pass inside the given circular arc AS, which does not exceed a quadrant, B being the lowest point of the circle. After the discussion of the problem enunciated in Art. 149 it will be sufficient to state the results of the present problem. The required curve consists of the arc of a cycloid having its cusp at A and touching the circle at some point R, and the por- tion EB of the circle. As the cycloid is outside the circle at R twice TR is greater than OR, and therefore TR is greater than OT. Hence for the point R we have y greater than b, and thus P~ a Sydx is necessarily positive for the part RB of the path, J2r* since y is positive. If AB is an arc of 90 there is no cycloidal portion, and the required curve consists entirely of the circular quadrant. 156. The problems of Arts. 149 and 155 include as special cases a problem proposed by the late Dr Whewell in the Smith Prize Papers for 1846. His enunciation was as follows : Prove that an arc of a circle from the lowest point which does not exceed 60 is a curve of quicker descent than any other curve which can be drawn within the same arc; and that the arc of 90 is a curve of quicker descent than any other curve which can be drawn without the same arc. BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 139 It would be interesting to know how the distinguished philo- sopher treated the problem himself. It will be seen that in our investigation we obtain for the circular arc this expression shews at once that the results enunciated are true with respect to such curves as differ infinitesimally from the given circular arc ; but it would still remain to shew that the results are true for curves which differ to a finite extent from the given cir- cular arc. The investigation which we have supplied establishes the results completely, including them in fact in wider statements. 157. Some extension may be given to the problem of Art. 149. Required the curve of quickest descent from a fixed point A to a fixed point J5, with the condition that the path is not to pass outside a certain boundary A CB composed of a circular arc A C, not exceeding a quadrant, and the straight line CB, which is the tangent to the arc at C, the lowest point of the circle. If A G is not greater than 60, the boundary is the required curve. If A C is greater than 60, the required curve consists of a part of A G beginning at A ; then an arc of a cycloid which touches AG at the point of departure, and either passes through B or touches BC : if the arc of the cycloid touches BC, the last portion of the required curve is a portion of BC. It depends on the length of BC whether the cycloid passes through B or touches B C. 140 BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. It will be seen that, taking u as in Art. 149, we have for the part of the boundary which coincides with BG this is positive, since By is here necessarily negative. 158. Some extension may also be given to the problem of Art. 155. Kequired the curve of quickest descent from a fixed point A to a fixed point B, with the condition that the path is not to pass inside a certain boundary composed of a circular arc BG, not ex- ceeding a quadrant, B being the lowest point of the circle, and the straight line GA which is a tangent to the circle at G. The required curve consists in general of an arc of a cycloid having its cusp at A, and touching BG', and of the arc of the circle from the point of contact to B. If BC is a quadrant, however, the boundary itself is the required curve. 7~P The expression iV -^- of Art. 149 reduces for the straight line AC to -- , ; and as by is here essentially positive, 2 the integral / a , ____ is negative: thus AC cannot form } 2yVl+/? 2 BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 141 part of the required curve. If, however, A is vertical we have p infinite, so that the integral just written may be considered to vanish : this agrees with our statement that AC is now part of the required curve. But it would be prudent in this case to choose polar co-ordinates, so as to avoid the infinite value of p. Take as the initial line the horizontal line through A ; then we have to find a minimum value of (r sin 6) * Call this u ; then where N=~ then P= Now take for the equation to the vertical straight line r cos = c ; dr c sin 6 a (. (dr\*\ * r Hence we shall find that cos 6 (r sin 0) 2 cos (r sin jj)* ' , r N-- and from these values it will follow that Thus the part of &u which corresponds to the vertical straight line vanishes. 142 BRACHISTOCHRONE UNDER THE ACTION OF GRAVITY. 159. Examples involving discontinuity connected with the brachistochrone might easily be multiplied : we will give another. Required the line of quickest descent from the fixed point A to the fixed point B, under the condition that the tangent to the path shall not make with the horizon an angle greater than a given angle ot. The following will be the solution in general : AC is a straight line inclined to the horizon at an angle a ; and CB is an arc of a cycloid which has its cusp in the horizontal line through A, and touches A G at C-. The part of Su which corresponds to AC will be as in Art. 158, f fy'(p)} Sydx The term of the first order which is outside the integral sign vanishes, since the limits are fixed. Then as usual we must have i/r (x) <$ (p) = G, a constant. Thus $u reduces to Consider the equation 4/(p) = C ........................ (i); this gives p in terms of c, from which y must be obtained. There will thus be two constants which must be determined by making the curve pass through the given extreme points. 102 148 PROBLEM OF LEAST ACTION. It is possible that in some cases a solution would be found by taking (7=0 and 0' ( p) = ; the solution would consist of one or more straight lines, corresponding to the various values of p. This case would resemble that of Art. 6. Suppose that when p is infinite <' ( p) has a finite value b ; then is a singular solution or a particular solution of (1). For (2) gives constant values for x ; and a constant value of x corresponds to p = infinity. In this case however on account of the infinite value of p it would be prudent to give another investigation, by polar co- ordinates for instance. However it appears that we may expect cases of discontinuity ; and we will now take a particular example. 164. Let u = U(x + a] (1 +/) dx. Then (1) becomes where for symmetry we put fjc instead of C. Thus p 2 = - - - ; x + a-c' therefore y- C' = + 2Vc (x + a - c) .... ............. (4), where C" is a constant. And as .. ^ ^ is unity, when^? is infinite we have a c for a singular solution of (3). Suppose the origin is taken at one of the fixed points ; then from (4) PROBLEM OF LEAST ACTION. 149 Thus (4) becomes y 2 Vc (a - c) = 2 Vc (x + a - c) (5). This equation represents two parabolas if c (a c) is positive. Let (h, k) denote the other fixed point through which the required curve is to pass : then Tc 2 Vc (a - c) = 2 Vc (h + a - c), so that k* 4>k Vc (a c) = 4cA ; therefore (& 2 -4cA) 2 = 16& 2 c (a - c) (6); therefore 16c 2 (A 2 + F) -8c (h + 2a) & 2 + & 4 = (7). From (7) we have two real values of c provided (h + 2a) 2 is greater than & 2 + A 2 , that is provided k 9 is less than 4a (a + h). It is plain from (6) that these values of c make c(a c) positive. Hence we obtain two parabolas provided that & 2 is less than 4a (a + h). 165. The example we are now discussing may be enunciated thus : determine the path of a particle for which the action I vds is a minimum between fixed points, if the velocity v at any point is that due to a fall from the straight line x + a = ; the axis of x being vertically downwards. If a particle is projected from a point with a given velocity, and acted on by gravity, we know that it will describe a parabola ; and if the particle is to pass through a second fixed point so long as this point lies within a certain boundary there are two parabolas either of which may be taken. Hence we conclude that our two parabolas are the two which might be described by a particle under the action of gravity if it started with the proper velocity. The equation k 2 = 4a (a + h) determines the boundary within which the second fixed point must be situated in order that the 150 PROBLEM OF LEAST ACTION. projectile starting with the proper velocity may reach it ; this boundary is a parabola ; we shall call it the bounding parabola. The value of $u, by Art. 163, is now If x increases algebraically constantly throughout the integra- tion this expression is certainly positive. If x diminishes alge- braically during part of the integration, then during that part we have to take (1 +p*)* and (1 +^> 2 )^ as negative ; so that the above expression is still positive. If then the arc of the parabola which we have to consider does not include the vertex, we may feel certain that we have a minimum. But if it does include the vertex, since at that point p is infinite, we cannot be certain that we have a minimum : the change of sign of (1 +p^)* at the vertex also would render our process suspicious. 166. Of course when the second fixed point is beyond the bounding parabola neither of the two parabolic paths is possible : so that some other minimum path must exist for this case, and may exist even if one or both of the parabolas is also a minimum. We have then to discover this other solution ; and also to settle the doubtful point noticed in the preceding Article as to whether a parabolic path is a minimum. 167. We shall first investigate whether a solution can be obtained by combining an arc of a parabola with the straight line defined by x + a = c: for we have already seen in Art. 164 that the last equation furnishes a singular solution of the differential equation (3) of that Article. It is essential that c in (3) should retain the same value at a point of discontinuity in order that the part of Su which is outside the sign of integration may vanish. Hence p must not undergo any abrupt change at a point of discontinuity. Thus if such a solution as we are now examining can exist, it must consist of a parabolic arc and the tangent at the vertex produced. PROBLEM OF LEAST ACTION. 151 Thus we have two cases to consider ; the parabolic arc being described before or after the straight line according as the point to be reached is higher or lower than the starting point. A is the starting point, B the other fixed point, G the vertex of the parabola. We have to determine whether this solution really gives a minimum. If we take the axis of x vertically downwards, as hitherto, the value of p is infinite at (7; and so Sp may be very great in the neighbourhood of C, and we cannot depend on the investigation of Art. 163. Let us then take the axis of y vertically downwards. Thus dx ; Thus it is obvious that p = gives neither a maximum nor a minimum for it leaves us with and we can of course make this positive or negative as we please. 152 PROBLEM OF LEAST ACTION. We have then still to seek a path of minimum action for the case in which the path to be reached is beyond the bounding parabola. 168. Now we have already in the course of our investigations seen that a solution of a problem in the Calculus of Variations may be furnished, at least in part, by some bounding line which by the 7) circumstances of the case cannot be transgressed : and we shall find that such is the case with the present problem. Let A be the starting point, B the point to be reached ; let CD be a horizontal straight line at a vertical distance a above A. Then CD is such a bounding line as we have supposed ; for above CD the expression for the velocity becomes impossible. We shall shew presently that the integral Ivds is a minimum for the path composed of the straight lines AC, CD and DB. Strictly speaking we cannot have a particle describing this path, because the velocity is zero at any point of CD. But we may suppose a curve drawn very close to this path; and the action along the curve will only differ infinitesimally from the value of the integral taken over the path. 169. We shall now shew that the integral Ivds really is a minimum for the path composed of A C, CD and DB. PROBLEM OF LEAST ACTION. 153 This will require some other investigation than those hitherto given in this Chapter in order to avoid infinite quantities. The equation (8) of Art. 167, for instance, is now unsatisfactory because y + a is zero along CD. Suppose Ac on the curve equal in length to A C on the straight line. Divide AC and Ac into the same number of equal infinitesi- mal elements. Let PQ and pq be a corresponding pair ; so that Ap is equal in length to AP. Then P will be vertically higher than p, and so the velocity at P less than the velocity at p ; and therefore the action over PQ will be less than the action over pq. In this way we see that the whole action over A C is less than the whole action over Ac. Similarly if Bd is equal in length to BD the whole action over DB is less than the whole action over Db. And finally the action along CD is zero, and that along cd is not zero. Thus the action for the path made up of A C, CD and DB is certainly less than the action for the adjacent curve. 170. We will now return to the point which was left unsettled in Art. 165, and determine under what circumstances a parabolic arc is really a minimum. We will take the axis of y vertically downwards, and put the origin at the highest possible point ; that is at a point where the velocity must be zero. Thus we have 154 PROBLEM OF LEAST ACTION. Hence proceeding in the ordinary way to find the minimum, we see that we must have where c x and c 2 are constants. Then by Jacobi's theory, in Art. 23, the term of the second order in Su is where ^ 6j and b z being arbitrary constants. Thus , = where o> stands for -^ * We see from (9) that CD =p. Now w r e know it is essential for a maximum or a minimum that z should not vanish during the range of the integration. To make z vanish we require that If then p -- can take every value between oo and + oo we are sure that z will vanish during the range of the integration. If -- p does not take every value between oo and + oo we can select T 2 so that z shall not vanish. PROBLEM OF LEAST ACTION. 155 If the arc of the parabola extends from one end of any focal chord to the other end p ranges from oo to oo , and there is not a minimum. If the arc of the parabola is of greater extent, then of course there is not a minimum. If the arc is of less extent there is a minimum. That is, to ensure a minimum it is necessary and sufficient that the arc should subtend at the focus an angle less than two right angles. 171. Suppose that a projectile is to start from A with a given velocity, and to pass through B. Then we know that the focus of the path must be in the intersection of two circles, one described from the centre A, and the other from the centre B> Thus one parabola will have its focus above AB, and th other will have its focus below AB. The former parabola does not give a minimum action, the latter does. This is in harmony with a result we shall presently obtain, namely, that of two parabolic paths between the same points the action is always greater for the higher parabola than for the lower. And the example furnishes an excellent illustration of Jacobi's process. If an arc of a parabola be greater than would be cut off by any focal chord', suppose a portion of the arc cut off by a chord passing below the focus but very close to the focus. Then suppose this arc removed and replaced by another with the same chord, but having its focus as much below the chord as the focus of the original arc is above the chord. Thus we can get a path differing only innnitesimally from the former, but giving a less action than the former : therefore the former arc cannot be an arc of minimum action. 15G PKOBLEM OF LEAST ACTION. 172. Hence our final conclusion is the following : the discon- tinuous path composed of three straight lines is always a path of minimum action; and for points outside the bounding para- bola, as there is no other path of minimum action, this must be the path of least action. For points inside the bounding parabola there is also another path of minimum action, namely, the lower of the two parabolas which could be described by a projectile between the two points. Hence for points inside the bounding parabola, as there are two, and only two, paths of minimum action, one of them will be the path of least action. As we shall see presently some- times the continuous path is that of least action, and sometimes the discontinuous path ; one or the other being necessarily such. 173. We will now estimate the value of the action in various cases. In a parabolic path the velocity being that due to a fall from the directrix is V2r^, where r represents the radius vector from the focus. Suppose 4n the latus rectum; then r= - 3 ; therefore & d0 cos 2 cos 2 Thus the action is .f M J~73- COS =. Now we know that if a particle were to describe a parabola under a force in the focus of the absolute intensity //, the time of . n* [ d0 describing an arc is Jg. 2 Thus we see that the action in the parabolic arc with which we are concerned is equal to 2 V/* dr Put p for -50 , and let then we require the minimum value of u. By the usual theory of the subject, we obtain r* 2 av = a constant ; ....(1). PROBLEM OF LEAST ACTION. 161 The integral of this is known to be M (0-/3)~ where e and ft are constants; for this reduces the value of the left-hand member to the constant Ja (1 e'). This integral may easily be obtained by finding , from the differential equation, and changing the variable from r to the reciprocal of r. For denoting the constant by *Jb we get Put - for r : thus u (^\-(2u--} l -u^ \dO) ( a) b therefore by differentiation This is immediately integrable. Now (2) is the equation to an ellipse, since the constant e is less than unity ; this ellipse has its focus at the centre of force, and has 2a for its major axis. 180. As the ellipse is to pass through a second given point, we have now to determine if such an ellipse can always be drawn so as to fulfil all the conditions stated. Since the major axis is 2a, the distance of the second focus from the starting point is 2a r 4 ; so that this focus lies on a circle described from the starting point as centre with the radius 2a rj. In like manner this -focus of the required ellipse lies on a circle described from the second fixed point as centre with radius 2a r 2 where r 2 is the distance of the second fixed point from the centre of force. If these two circles intersect, there are two positions for ihe second focus, and two ellipses can be obtained ; if the circles touch, one ellipse can be obtained ; if the circles do not intersect, there is no such ellipse as we require. 11 162 PKOBLEM OF LEAST ACTION. 181. The boundary of the space within which the second fixed point must be situated in order that an elliptic arc may connect the two fixed points is an ellipse, as we shall now shew. Let be the centre of force, A the fixed starting point, P any point on the boundary : then we must have - OP\ . therefore OP+ AP=4>a- OA. Thus the locus of P is an ellipse of which and A are the foci, and the major axis is 4>a OA : the excentricity is the ratio of OA to the major axis, that is 7-- . 182. Now guided by the analogy of our preceding investi- gations, we may anticipate the following results for our problem of minimum action. Denote the fixed points by A and B. There is always a discontinuous minimum solution which is thus obtained : the radii vectores to A and to B must be produced until they meet the circle described with as centre and the radius 2a; the produced parts of the radii vectores together with the arc of the circle which they intercept constitute the solution. It may be shewn to be a true minimum by the process of Art. 169. If the second fixed point is outside the bounding ellipse this is the only minimum, and therefore gives the path of least action. PROBLEM OF LEAST ACTION. 163 If the second fixed point is inside the bounding ellipse, two elliptic arcs can be drawn between the two fixed points ; only one of these, however, is a curve of minimum action, namely, that which has both its foci on the same side of the straight line which joins the two fixed points : this can be shewn by Jacobi's method as will presently appear. This example of his method was indicated by Jacobi himself. [See Todhunter's History of the Calculus of Variations, page 251.] Thus, if the second fixed point is inside the bounding ellipse, there are two paths of minimum action, one continuous and the other discontinuous; in some cases one of the two is the path of least action, and in some cases the other is : one of the two being necessarily the path of least action. 183. We will now determine by Jacobi's method which of the two elliptic arcs gives a path of minimum action. We denote by the centre of force ; then being the origin the equation to an ellipse is a (1 - e 2 ) A* ^^ 1 + e cos (0 - j3) ' Let 8 denote the other focus. It is obvious that for one ellipse S and are on the same side of AB, and for the other ellipse 8 and are on opposite sides of AB\ in the former case the arc with which we are concerned subtends at 8 an angle less than two right angles, and in the latter case an angle greater than two right angles. We shall shew that in the former case the path is one of minimum action, but not in the latter case. The quantity denoted by z in Art. 25 here stands for , dr , dr where ^ and 2 are arbitrary constants. AT dr dr OW ^7/Q ~ ~~ e ~JA ~ ~~ 6 P' 112 164 PROBLEM OF LEAST ACTION. To find -j- we have = 1+6008(0-0); f 2ae a(le*)dr . Q _ therefore ---- ^ - -r- cos (6 - 8) r r- de dr y 2 -ar(l+e 2 ) hence j- = - ,., v T. ' . de ae (1 e ) Thus z = (7 t jr 2 - ar (1 + e 2 ) + rapi , where C l and m are arbitrary constants. Hence we must examine the range of values of which P is susceptible. We shall find that if the elliptic arc subtends at S an angle of two right angles, the expression is susceptible of all values between oo and + oo ; and of course this will therefore be 'true if the angle is greater than two right angles : hence in these cases there will not be a minimum. But if the elliptic arc subtends at S an angle less than two right angles the expression is not susceptible of all values, and so there will be a minimum. It is obvious that - is infinite and changes sign at a vertex of the ellipse ; hence we shall establish our point if we shew that at the two ends of a chord through the focus this expression has the same value and the same sign ; for then the expression, as it begins with a certain value, changes sign as it passes through infinity, and finally returns to its original value, will have passed through all values. PROBLEM OF LEAST ACTION. 165 Now the value varies as r - a (1 + e 2 ) r sin (0 - ) ' Put 2a p for r, and p sin $ for r sin (0 /3) ; so that /o is the radius vector to the focus S, and is the angle which this radius vector makes with the major axis of the ellipse. Hence the expression becomes a (I- e z ) - p Now sin has numerically the same value at the two ends of the chord through S, but with opposite signs ; let p l and p 2 denote the two values of p. Then we have only to shew that that is a (1 - e 2 ) (- + -} = 2 ; Vi P*' and this is obviously true. 184. The action through any arc of the ellipse described under a force at is known to vary as the time of describing the same arc under a force at S\ the theorem of Art. 173 is indeed a particular case of this. Thus we might express the action as in Art. 173 by using the theorem for an ellipse, which is analogous to Lambert's theorem for a parabola. 185. It is easy to illustrate discontinuous solutions still further, by an example like that of Art. 178. Required a path 166 PROBLEM OF LEAST ACTION. of minimum action between the two fixed points A and B with the condition that the path shall lie entirely between two circles, one described with centre and radius OA, the other described with centre and radius OB. It is possible that an arc of an ellipse alone will satisfy the conditions. If not, the path must be composed of an arc of an ellipse having one extremity at one fixed point, and touched at its vertex by an arc of a circle described from as centre with a radius equal to the distance from of the other fixed point. [This will be sufficiently obvious from what has been already frequently stated. The path can consist of nothing but some combination of an arc of an ellipse with an arc of the prescribed boundary. It is necessary that the two arcs should touch at the common point in order that the term in the variation which is free from the integral sign may vanish. Thus the common point must be a vertex of the ellipse. Let u be taken as in Art. 179. Then it will be found that for /(a _ r\ fa -p -dO. */2a-rra This is positive. For suppose that the arc of the circle is part of the larger circle ; then Sr is necessarily negative along the circular arc. And a r is negative. For, by hypothesis, the elliptic solution which would be applicable if there were no condition imposed is now inapplicable ; that is, this ellipse would cross the boundary. But by Art. 183 this ellipse has both its foci on the same side of AB : and this makes the distance from of the more remote of the two points A and B greater than a ; that is, r is greater than a. In like manner if the arc of the circle is part of the smaller circle Su is positive ; for then Br and a r are both positive.] L I P> II A R V | UNIVEKSIT V OF CALIFORNIA CHAPTEK IX. SOLIDS OF MINIMUM RESISTANCE. 186. To find the form of a solid of revolution which experi- ences a minimum resistance when it moves through a fluid in the direction of the axis of revolution. I need scarcely say that the interest of this problem arises from its historical connexion with many illustrious names, including that of Newton ; and is thus quite independent of the amount of practical value of the results, and of the trustworthiness of the ordinary theory of the resistance of fluids. Take the axis of x as that of revolution ; then by the ordinary principles we require a minimum of Hence, in the usual way, we obtain * 3/>*(l+/)-2p 4 = UP - P + constant ; therefore - W = constant ................. (1). Denote the constant by - 2c t ; then we have 168 SOLIDS OF MINIMUM RESISTANCE. Hence, by differentiation, -3) dp p= ^- -^ -p-f- p r dy Thus ~~ is positive or negative according as p z is greater or dy less than 3 ; for we may take c t to be positive : in the former case the generating curve is convex to the axis of x, and in the latter case concave. Now denoting ~-^ by (p), we find that Hence by the general investigation given in Art. 26 we have for the term of the second order in &u It follows at once that there will be no minimum if p* is greater than 3 within the range of the integration. But if p* is less than 3 throughout this range the curve is concave to the axis of x as we have already seen : thus y" is negative, and we have necessarily a minimum. ? + + c* (3), where c 2 is an arbitrary constant. Let us now suppose that the generating curve is to terminate at fixed points ; so that the surface is in fact to be a zone of a surface of revolution. From (2) and (3) theoretically we must eliminate p, and thus obtain an equation between x and y and SOLIDS OF MINIMUM RESISTANCE. 169 the constants c t and c a . Then having given the co-ordinates of the two fixed points we have two equations for determining the constants. Thus theoretically all is satisfactory. But there are some important remarks to make on the solution. 187. We must be careful not to speak of our solution as giving the solid of least resistance. Legendre in fact pointed out that by taking a zigzag line for our generating curve we might make the resistance as small as we please. [See Todhunter's History of the Calculus of Variations, page 229.] Our solution gives us only a solid of minimum resistance ; by which we mean that if we pass from our generating curve to another by giving to y and p infinitesimal changes, we shall obtain a solid of greater resistance. But our investigation does not allow us to pass from our curve joining the fixed points A and B to the zigzag line ; because although the change in y might be infini- tesimal throughout, yet the change in p would not be such. It should be observed that Legendre's zigzag line may be supposed to be suggested by the fundamental equation (1) of Art. 186 ; inasmuch as p = is a solution of that equation, namely, when the constant is zero. Legendre himself does not make this remark, which is however important ; for otherwise we might be left with a feeling of general distrust, and the expectation that solutions will present themselves in an arbitrary manner without the warning of the fundamental equation. 170 SOLIDS OF MINIMUM RESISTANCE. 188. Suppose however we add to our problem the condition that the generating curve is to have p always with the same sign. It is obvious that there must be some curve with this condition which generates the solid of least resistance ; and our investigation assures us that it can be no other than the curve which we have obtained ; unless it be some discontinuous line to which we shall presently advert. Hence finally; if we do not impose the condition that p is to be of invariable sign, we can only say that our solution gives a minimum with respect to infinitesimal varia- tions of y and p ; but if we impose the condition, we may assert that our solution gives the solid of least resistance unless any discontinuous solution can be found. 189. But it is easy to see that the continuous solution which we have obtained cannot be universally applicable. Let A and B be the fixed points ; and suppose that the straight line drawn from A to B is inclined to the axis of x at an angle greater than 60 : then it is certain that we cannot draw a curve from A to B with the condition that p* shall always be less than 3. We must therefore seek for another solution. Now if we limit p to have always the same sign, we do in effect determine that our curve shall not fall outside the rectangle SOLIDS OF MINIMUM RESISTANCE. 171 which has AB for a diagonal, and has two of its sides parallel to the direction of motion. Therefore we naturally proceed to enquire if the boundary thus assigned does not itself in part constitute the generating curve. Suppose the axis of y to pass through A. We propose to seek for a solution composed of AC, which is part of this axis, and a curve CB which satisfies equation (2). 190. Let OG y^ and OA = &; let the abscissa of B be a, and the ordinate b. Then we seek for a minimum of Then find Su, and make the part of &u which is of the first order and under the integral sign vanish : when this is done, the remaining term of the first order in Bu is the subscript indicating that the values correspond to the point C. To make this term vanish, we must have which leads to ^ = 1. Thus the curve must meet the axis of y at an angle of 45. Now consider the part of Bu which is of the second order. From the term y 2 in u there arises (S# ) 2 . From the other term in u, remembering that y Q is not constant, we have by the general investigation of Art. 26, l>) \y T where < (p) stands for 5 . 172 SOLIDS OF MINIMUM KESISTANCE. Thus we have (p) = 1 when p = \. Hence Su reduces to which is essentially positive. 191. It only remains to enquire if real values can be found for the constants c x and c 2 which occur in our discontinuous solution. Let -CT denote the value of p at B ; then since p = 1 at A, we have, by equations (2) and (3), from these we have to determine C , c,, y and r. Substitute the value of c 2 from the second of these equations in the last ; then we have y =40i .................... . ............................ (4), ( 6) - From (5) and (6) we determine tsr ; for by division we have tzr 3 log 'cr + 'S7+-: -- T OT 3 4sT 4 since the expression on the second side of this equation would change from infinity to zero, as -DT changed from 1 to 0, a real value of -or to satisfy the equation must exist. Then from (5) we find c t , and from (4) we find y . SOLIDS OF MINIMUM RESISTANCE. 173 192. In order, however, that this solution may hold, we must have y greater than k, that is 4c, greater than k. It is not in our power to give a simple expression for the relation which must hold between a, b, and k at the limit of the possibility of the solution. We can however shew that if b k is greater than a the possibility is ensured. For from (5) and (6) we have b-a = c,z (7), where z stands for thus * Hence we see that z 4 when -sr = 1, and that z diminishes as tzr diminishes from the value unity. It follows then from (7) that 4c 4 is never less than b a ; and as b a is by hypothesis greater than k, we have 4c t greater than k. Thus the solution which we have obtained is certainly ad- missible in some cases in which the continuous solution is not admissible, namely, the cases in which b k is greater than a\/3. 193. The next question is whether any other solution can be found. The ordinary theory of the Calculus of Variations assures us that there can be no continuous solution except that which we first obtained, or, in other words, that there is no other solution in which By is susceptible of either sign. Thus there can be no other solution than the continuous solution, and a solution or solutions composed in part of the boundary to which we are by supposition restricted. We have already investigated one such discontinuous solution. Two other cases present them- selves for examination. We may try if the generating line can be composed in part of a straight line parallel to the axis of revolution drawn through the more remote of the two fixed points. But 174 SOLIDS OF MINIMUM RESISTANCE. it may be ascertained immediately that no solution of this kind exists. For the upper limit of integration will now not be fixed ; denote this limit by a? r Then we should require a minimum of 2/0 and hence in the variation there arises a term which does not vanish. The only remaining case for examination is that in which the generating line consists of the curve AC, and the portion CB of the ordinate at B. Let y l denote the ordinate of C. Proceed as in the former case. We now seek a minimum of The term of the first order in the variation outside the integral sign will now be To make this vanish we must have p l 1, so that the curve must now meet tjhe ordinate of B at an angle of 45. SOLIDS OF MINIMUM EESISTANCE. 175 The term of the second order in the variation arising from y* is cancelled by a similar term arising from the variation, 1 f a vv* of 2 I :p - dx ; and we are finally left with the same essentially "o * ~t~ P positive value of the term of the second order in the variation as we had in Art. 190. 194 To determine the constants c t and c 2 we have the fol- lowing relations, where -cr now denotes the value of^? at A. 07' And -cr must lie between 1 and \/3 ; also 4Cj must be less than b. By eliminating c 2 and c, we arrive at the following equation : a OT 3 (7 13 As OT varies from 1 to V^ the right-hand member of this . , 3 V3 /4 Iog3\ . a expression varies from to -y^- ( 1 1 , so that if T lies between these limits a real value of or can be found to satisfy the equation. But this solution only gives a solid of minimum resistance, and not the solid of least resistance, as will appear from the next Article. 195. In Art. 190 we see that the tangent to the curvilinear part of the solution never makes with the axis of revolution an angle greater than 45. This result is in harmony with an interesting proposition given by the late R. L. Ellis ; see Quarterly Journal of Mathematics -, Vol. x. p. 122. It follows from Mr Ellis's proposition that our solids would not be solids of least resistance 176 SOLIDS OF MINIMUM RESISTANCE. even with the condition of having p always positive if the value of p were ever greater than unity. The proposition of R. L. Ellis is a generalization of one given by Newton : see the Principia, Book II. Prop. 34. Newton may be said to be the first person who treated a problem of the Calculus of Variations; and that problem involved a discontinuous solution. 196. Let us now finally sum up the results we have obtained with respect to the continuous solution and the two discontinuous solutions. By virtue of Legendre's remark already noticed none of these solutions gives us a solid of least resistance. Every one of these solutions when it really exists gives us a solid of minimum resistance ; that is, any admissible variation in the form of the generating curve would increase the resistance ; by an admissible variation, we mean that $y and $p must be always infinitesimal, and that our curve is to be comprised be- tween the extreme ordinates, so that no variation is required for x. If we impose the condition that p is to be of invariable sign, then in every case one of our solutions gives the solid of least resistance ; namely, the solution which exists, supposing that only one exists, and that for which the resistance is least when more than one exists. 197. The discontinuity which occurs in the preceding inves- tigations may be considered to arise from the conditions which we impose, namely, explicitly that p is to be of invariable sign, and implicitly that the curve is to be comprised between the extreme ordinates. The fundamental relation of Art. 186, namely constant, may be considered to be satisfied in a certain sense by p = oo , and this may suggest a straight line parallel to the axis of y as forming a part of the solution. But unless the conditions be imposed we shall not be able to shew that we have really a minimum. SOLIDS OF MINIMUM RESISTANCE. 177 For example, if in Art. 189 we were at liberty to extend to the left of the axis of y we might replace A C by a boundary which would give a less resistance. 198. A particular case of the preceding problem may deserve special notice, namely, that in which one of the fixed points is on the axis of revolution. Generally suppose we require a maximum or minimum of y$ (p) dx. By the ordinary method we obtain y{$(p) ~P& (p)} = constant. If one of the extreme points is on the axis of x, and (p) pcf) (p) is never infinite, the constant must be zero. In the present case we thus obtain This will not furnish us with any minimum, except we regard as such Legendre's zigzag line which may be supposed to corre- spond to p = 0. We may consider that we obtain a maximum by combining y = with p = oo , that is, by taking a portion of the axis of x and the ordinate of the second fixed point ; for thus we obviously obtain the greatest value. If we impose the condition that p is to be always of one sign, we shall obtain a minimum like that of Arts. 189... 191 ; the solution is always applicable, for we shall not now require as in Art. 192 that 4c x shall be greater than an assigned positive quantity. And as no other minimum presents itself, we infer that we have the figure of least resistance, under the assigned condition. 199. The following elementary problem will serve to illustrate the result which we obtained in the discontinuous solution of Art. 190, that the curve meets the initial ordinate at an angle of 45. 12 178 SOLIDS OF MINIMUM RESISTANCE. A and B are fixed points, P is a variable point on the ordinate at A : it is required to determine the position of P, so that the resistance on the surface generated by the revolution of the straight lines AP and PB may be the least possible. Let OPy ; let OA = & ; let b be the ordinate of B and a the abscissa. Let 9 be the inclination of PB to the axis of revolution ; then b = The resistance varies as y* & 2 4- (b z t/ 2 ) sin 2 6, that is as 2/ 2 cos 2 + 2 sin 2 0-& a , that is as (b cos - a sin 0) 2 + V sin 2 - & 2 , that is as b 9 &* 2a& sin cos 6 + a 2 sin 8 0, that is as 2 2 _ #> + _ al s i n 20 - * cos that is as ,0/1 A cos (20 - a), where cos a The resistance is therefore least when 20 = a. If a be very small this gives very nearly 20 = ^ We m'ay observe that if is greater than j- our expression for the resistance increases with 0. SOLIDS OF MINIMUM RESISTANCE. 179 200. Another elementary investigation may also be usefully supplied here. Suppose PQ an indefinitely small arc of a curve; PR and RQ any indefinitely small arbitrary straight lines: required an expression for the difference between the resistance on the strip of surface generated by the revolution of PQ and the sum of the resistances on the strips of surface generated by the revolution of PR and RQ. Let x and y be the co-ordinates of P; x + dx and y + dy the co-ordinates of Q. Let PR = v, RQ = w. Let a be the inclination of PR to the axis of #, and ft the inclination of RQ to the axis of x. Let PQ = ds. Now omitting a certain factor in the usual notation, which is constant, the resistance on the strip of surface generated by the revolution of PQ will be ultimately 2y (-?} dy ; the resistance on the strip of surface generated by the revolution of PR will be ultimately 2yv sin 3 a ; and the resistance on the strip of surface generated by the revolution of RQ will be ultimately 2ywsm*p. Hence we require the value of 122 2y dy - v sin 3 a - w sin' 180 SOLIDS OF MINIMUM RESISTANCE. But v sin a 4- w sin ft = dy, and v cos a + w cos ft = dx ; , i f dy cos 8 dx sin 8 therefore v = -* =-? ~ - , sm (a /3) , dx sin a. dy cos a and w = ; %r . sm (a p) Thus our expression becomes 9 (dycos/3 y " sin(a-/3) and this = 2y \ \-j-\ dy + dx sin a sin ft sin (a + ft) . ( ^_o\ t cos ft sm a (1 - cos 2 a) cos a sin ft (1 - cos 2 /3)] I f ffJu\ 2 2# ] f^J ^ + ^ sin a sin /3 sin (a + ft) dy + dy cos a cos ft cos (a + /3) [ . By putting this into factors we obtain ~r- 2 (dy cos a dx sin a) (dy cos ft dx sin /3) f cfo/ cos 7 + da? sin 7) ; where 7 stands for a + ft. IT 7T For a particular case, suppose a = ^ an d = -j ; then our expression becomes ydx(dy dx)* ds* and this is positive. This particular case and the next particular case are given in the Quarterly Journal of Mathematics, Vol. x. page 122. SOLIDS OF MINIMUM RESISTANCE. 181 This shews that if we have a curve such that the tangent is everywhere inclined to the axis of x at an angle greater than 45, we can diminish the resistance on the solid generated, by passing from the curve to the notched boundary where the straight lines are alternately inclined at 90 and 45 to the axis. For another particular case of the general result, let PR be inclined to the axis of x at 45 and EQ at 90; then we shall find that the sum of the resistances corresponding to PQ and QK exceeds the resistance corresponding to PR by ydx (dy dx}' ds* and this is positive. 182 SOLIDS OF MINIMUM RESISTANCE. 201. Now let AP be any arc of a curve, such that the tan- gent PT at P is inclined to the axis of x at 45 ; let A T be inclined to the axis of x at 90. We proceed to compare the resistance on the surface generated by the revolution of AP with the resistance on the surface generated by the revolution of A T and TP. Let OAy^ OT a\ let h and k be the co-ordinates of P. We shall now estimate the resistances ; we omit, as before, a certain constant factor. We imagine a series of zigzags drawn like that of the first particular case of Art. 200. The resistance on the surface corresponding to AT and TP The resistance on the surface which would correspond to the vertical parts of the zigzags along AP SOLIDS OF MINIMUM RESISTANCE. 183 = 2 2/o > - * - twice the area OAPM. . The resistance on the surface which would correspond to the inclined parts of the zigzags along AP = 2 ( y \ cfo = the area OAPM. J a * Thus the resistance on the surface which would correspond to the whole of the zigzags = tf -y*- the area OAPM. This exceeds the resistance on the surface corresponding to AT and TPby | (7c 2 - a 2 ) - the area OAPM, Zi that is, by the area A TP\ for i (k + a) (k-a) is | (OT+PM) OM, and is therefore equal to the area OTPM. By the first particular case of Art. 200, the resistance on the surface generated by the revolution of the curve AP exceeds the resistance on the surface which would be generated by the revolution of the zigzags by I ^-^ , where the integral J ds extends throughout the curve. Thus finally the resistance on the surface generated by the revolution of the curve AP exceeds the resistance on the surface generated by the revolution of AT and TP by fr-**)^ the area ATP. SOLIDS OF MINIMUM RESISTANCE. 202. Next suppose the curve and the tangent produced beyond P\ let BS be inclined to the axis of x at an angle of 90 : we shall compare the resistance on the surface generated by the revolution of PS with the resistance on the surface generated by the revolution of PB and BS. Let x lt y 1 be the co-ordinates of B\ let SC'b. T O We imagine a series of zigzags drawn like that of the second particular case of Art. 200. The resistance on the surface corresponding to PS diminished by the resistance on the surface corresponding to BS The resistance on the surface which would correspond to the vertical parts of the zigzags along PB y (dx - dy) = twice the area MPB C - (y* - V). The resistance on the surface which would correspond to the inclined parts of the zigzags along PB = the area MPBG. SOLIDS OF MINIMUM RESISTANCE. 185 Therefore the resistance on the surface which would cor- respond to the inclined parts of these zigzags diminished by the resistance on the surface which would correspond to the vertical parts = y* -tf- the area MPBC. This exceeds the resistance which corresponds to the difference of PS and SB by | (j* _ jf) - the area MPBC, that is, by the area of PBS. By the second particular case of Art. 200 the resistance on the surface generated by the revolution of the curve PB exceeds the resistance on the surface which would correspond to the difference of the vertical and inclined parts of the zigzags by y dx (dy dx)* Thus finally the resistance on the surface . generated by the revolution of the curve BP exceeds the resistance which corre- sponds to the difference of PS and SB by { y dx (dy dx) z . T>aT> - - r^ - L + the area PSB. J or Or, which is the same thing, the resistance on the surface generated by the revolution of the curve PB, together with the resistance on the surface generated by the revolution of BS, exceed the resistaDce on the surface generated by the revolution of PS by C J ds* The investigations of this and the preceding Article are omitted in the paper of the Quarterly Journal of Mathematics, to which I have already referred, although they are necessarily required there. 186 SOLIDS OF MINIMUM RESISTANCE. 203. The proposition given by R. L. Ellis, to which I referred in Art. 195, is the following : Take the diagram of Art. 202 ; let AB be an arc of a curve ; let T8 be a tangent at P, inclined at an angle of 45 to the axis of x let TA and SB be inclined at an angle of 90 to the axis of x. Then if the figure revolve round the axis of x, the resistance on the surface generated by ST and TA is less than the resistance on the surface generated by SB and BPA. The demonstration is contained in Arts. 201 and 202. Mr Ellis established his theorem geometrically. An analytical investigation is given in the Quarterly Journal of Mathematics which is unsatisfactory; for it confounds the resistance on the surface corresponding to AT and TP with the resistance on the surface which would correspond to the zigzags along AP. But these two amounts of resistance are not equal ; the latter is the greater, as we have seen in Art. 201. 204. The proposition given by R L. Ellis may also be esta- blished in another manner, which indeed resembles his own. We will confine ourselves to the part PA of the curve, as the remarks made can be easily applied with suitable modifications to the part PB. The curve may be supposed to be generated by the perpetual intersections of straight lines. Suppose QE and RF to be two consecutive straight lines. Then I shall shew that the resistance on the surface corresponding to FE and EQ is less than the re- sistance on the surface corresponding to FR and R Q ; or, which is the same thing, the resistance on the surface corresponding to FE and ER is less than the resistance on the surface corresponding to FR. It is obvious that if this be true we may pass by a series of changes at every one of which the resistance is diminished, from the resistance on the surface corresponding to the AP of Art. 202 to the resistance on the surface corresponding to the AT and TP. Now the fact that the resistance on the surface corresponding to FE and ER is less than the resistance on the surface corre- SOLIDS OF MINIMUM RESISTANCE. 187 spending to FR is obvious by the last line of Art. 199. It may be established by a formal use of the method of variations thus : Let y refer to points on FR, and let Sy denote the variation by which we pass from FR to ER. Let u I *p- 2 ; so that %TTU measures the resistance on the surface corresponding to FR. The change in passing from FR to FE and ER is therefore Since ^> is constant we see that Su reduces to 188 SOLIDS OF MINIMUM RESISTANCE. Hence 27n/ Sy 4- ' This is certainly negative if p is not less than unity. 205. We may briefly advert to the problem of finding a solid of maximum resistance. In this case we should have the same fundamental equation as in Art. 186. The solid of greatest resistance will correspond to the solution given by combining p = with p = oo . The generating curve may be supposed to consist of a straight line through A parallel to the axis of x, and the part of the ordinate of B cut off by this straight line. The same amount of resistance will of course be obtained by a line composed of a series of steps. Perhaps besides this solution which gives the greatest resist- ance a solution might be found to give a maximum resistance for admissible variations. Jacobi's theory shews that the value of p* in the curve corresponding to (1) of Art. 186 must in that case never be less than 3. If we require the generating curve to be of given length, there will certainly be some solution for which the resistance is greatest. The equation which will be obtained in the next Article of course applies here. 206. Suppose it were required to determine the solid of revo- lution of least resistance on the supposition that the generating curve is terminated at two fixed points, and has a given length. By the usual theory we have to seek a minimum of where c is a constant. SOLIDS OF MINIMUM RESISTANCE. 189 This leads to then x may be expressed in terms of p by the relation ./& />' But the expressions are too complicated to be of any service. We may however be sure that there must now be some solid of least resistance under the condition of given length of the generating curve. Suppose the given length happened to be exactly the length of the curve in Art. 190 ; then that curve would give a minimum solution of the present problem. If we impose also the condition that p is to be always of one sign, then in the case in which the solution of Art. 190 is the only solution of the problem without the condition of given length, it will be the only solution of the present problem if the given length happens to coincide with the length thus obtained. Hence we have a dis- continuous solution of the present problem in certain cases. 207. We proceed to another variety of the problem of a solid of minimum resistance. A solid of revolution is to be formed on a given base, so as to have a given surface and to experience a minimum resistance when it moves through a fluid in the direction of its axis. Take the axis of x for that of revolution, and make y the independent variable. We have then to find a minimum of dx where r stands for -=- and c is a constant. dy Denote the integral by u ; then 190 SOLIDS OF MINIMUM RESISTANCE. The term of the first order is transformed in the usual way, and is made to vanish by the supposition yvr VJ a constant ; and this constant must be zero since the generating curve is supposed to meet the axis of x. We have then to form a solu- tion consisting of a straight line or straight lines, which are furnished by | 1. Thus if BC = b, and we draw BA so that the surface generated by the revolution of BA may have the given value, this conical surface satisfies the conditions for a minimum. For the term of the first order in Su vanishes, and the term of the second order /- ( becomes | J 2x , (S^) 2 %, which is positive. 208. If ED is equal to EA we may take the discontinuous locus formed of BE and ED instead of that formed by BA. Thus if we propose that the generating curve shall pass through a fixed point D on the axis we may suppose that our generating curve is composed of BA and AD, or of BE and ED : the surface and the resistance are the same in the two cases. SOLIDS OF MINIMUM RESISTANCE. 191 The discontinuity which exists when we take BE and ED for the generating line is like that of Art. 9, and arises in the same way : the equation here from which -cr is found gives us two values numerically equal but of opposite signs. It is obvious that we may increase the number of zigzags with- out changing either the area of the surface or the amount of the resistance ; and so the generating curve can always if we please be comprised between the ordinate at B and any other fixed ordinate. 209. We observe that the case OT = presents itself among those to be examined in Art. 208. This gives great variety to the figure which has the property of a minimum. For example, suppose DE parallel to the axis of y, and FE and DB parallel straight lines. Let the lengths of the lines be such that if FEDB revolve round the axis of x the surface generated has the given value. Then the figure thus formed has the minimum property ; that is, any figure obtained from this by an admissible variation, and generating an equal surface, will corre- spond to a greater resistance, provided the variation is not limited to the part DE alone. For the term of the first order in Bu vanishes, and the term of the second order vanishes for the part DE, and for the rest of the figure retains the form given to it in Art. 207; and this is positive whatever may be the value of or. 210. But such a figure as that in the preceding Article will not correspond to the least resistance. To shew this we have only to consider the following proposition : 192 SOLIDS OF MINIMUM RESISTANCE. A surface is generated by the revolution of a straight line PS round the axis of x j another surface is generated by the revolution O 3, # of the composite line PQ and QE; supposing the areas of the two surfaces equal, the resistance on the latter is the greater. Let PT=b, QT=r, PST=a, QET=ft. Since the surfaces are equal we have r z ,, f b 2 (1 - sin a) sin ft ; therefore r = sin a sin /3 . . . (1 sm p) sin a The resistance corresponding to PS varies as b 2 sin 2 a ; the resistance corresponding to PQE varies as r 2 sin 2 ft + b* r 2 : the latter = b 2 sin 2 a + b 2 (1 - sin 2 a) - r 2 (1 - sin 2 /3) . f . 2 sin^(l-sina)(l + sin/3)) = o sin a + o < 1 sin a i r \ sin a J = 6 2 sin 2 a H ^ . 'l 111 ^ J (1 -} sin a) sin a - (1 + sin ft) sin m sin ot I J = i 2 sin 2 a H ~s ( s in a sin yS) (1 + sin a -f- sin ft). It is obvious that this is greater than 6 2 sin 2 a. 211. Thus it is clear that we cannot obtain the solid of least resistance if we make any use of the solution -cr = 0. As to the solution y = this would correspond to -57 = 00, and the investigation which has been given is not very satisfactory in this case. But on changing the independent variable from y to x SOLIDS OF MINIMUM EESISTANCE. 193 we shall see that y = corresponds to a minimum. But this result is of no consequence, as the surface and the resistance both become zero when y = ; so that it makes no difference what portion of the axis of x is comprised in our solution. Hence it follows that to obtain the solid of least resistance we must take the solution of Art. 207. 212. Suppose in Art. 210 that Q is very near to P; still the conclusion holds that the resistance on PQ and QR is greater than that on PS. This may at first appear inconsistent with the statement of Art. 209, that the diagram there given corresponds to a minimum ; for it is clear that by such a change as consists in taking PS instead of PQ and QR the resistance is diminished. But it must be observed that such a change cannot be made by an admissible variation ; in passing from PQ to PS although the variation in x may be infinitesimal, that in VT will not be infini- tesimal. 213. Suppose in this problem that we require the generating curve to terminate at fixed points, and also impose the condition that shall never change sign ; the following is the solution : Let A and B denote the fixed points, A being on the axis of revolution. Draw BC and BF perpendicular to the axes. If the given area of the surface lies between that gene- rated by BG and that generated by BA, the required solution is made up of two par Is AD and DB. If the given area of the surface lies between that generated by BA and that generated by AF and FB, the required solution is made up of two parts AE 13 194 SOLIDS OF MINIMUM RESISTANCE. and EB. In the former case we in fact combine c 2 = and in the latter case we combine -cr = and We may observe that it can be immediately shewn that the area of the surface in the second case increases as AE increases. For take the general expression for &S given in Art. 84, and put q = ; thus we obtain for the variation produced by a change of a straight line BE to a slightly higher position the latter term is positive, the former term is less than which is the increment of the surface generated by AE. 214. Let us now briefly advert to the problem in which the resistance is required to be a maximum, the area of the surface being given as before. It is evident that the greatest resistance is obtained by supposing the surface generated by a straight line DBG at right angles to the axis of such a length that the area of the circle thus obtained may be equal to the given area. This result may be extracted too from the formulae ; the solution must be considered to be tn- = 0. Let CD be denoted by y l ; then we must remember that y^ is variable, and so to the SOLIDS OF MINIMUM EESISTANCE. value of &u in Art. 207 we must now add 195 that is v (1 + c since -cr = 0. To make this vanish we should require c 1 ; and then the term in Su of the second order is negative. If however the generating line is not allowed to have any ordinate greater than that of B we must take a different solution. Draw EG parallel to the axis of x and CD perpendicular to it ; let EG be of such length that the surface generated by the revolution of SO and CD may have the given area : then the solution may be considered to be made up of BC and CD. The part CD corresponds to r = ; the part CB corresponds to -BT .= oo , which with c = may be considered to satisfy the fundamental equation of Art. 207. It would be unsafe to rely upon the investigation on account of the infinite value of w ; but it is obvious that this figure gives us the greatest possible amount of resistance. 132 CHAPTER X. SOLID OF MINIMUM RESISTANCE WITH GIVEN VOLUME. 215. THE following interesting variety of the problem of the solid of minimum resistance has been recently considered : a solid of revolution is to be formed on a given base with a given volume so as to experience a minimum resistance when it moves through a fluid in the direction of its axis. See Philosophical Magazine for November, 1867. I borrow little more than the enunciation of the problem ; in fact the discussion which will no\v be given is almost entirely new: all that relates to the interpretation of the results, and to discontinuity, is of course here given for the first time. 216. Let Ox denote the axis of revolution, AB the generating curve, BG the radius of the given base. o Take the origin at any point of the axis; let 00= a, and T}j these may be considered known quantities. Let OA = X Q which is not known. SOLID OF MINIMUM RESISTANCE WITH GIVEN VOLUME. 197 By the usual theory we seek for a minimum of where X is a constant at present undetermined. By putting the variation of this to the first order equal to zero, we obtain in the usual way a constant. ' - The constant here must be zero since the curve meets the axis of x. Hence p* r or putting - for \, we have c Thus (1) determines the generating curve; we shall shew that this curve is a hypocycloid. Put p = tan ; then y c sin 3 cos < ; .1 f dlJ , n . o . . 4 \ ^ therefore = c (3 sm cos sin 9) -f ; ^ x aa? therefore -^ = c (3 cos 2 sin 2 0) sin 2 (/>, ^ and -=-7 = c (3 cos 2 ) sin <^> cos ) sin 6 = c sin 3(, ot^> where 5 denotes the arc of the curve ; therefore 5 = constant cos 3. o 198 SOLID OP MINIMUM RESISTANCE Hence the curve is a hypocycloid in which the radius of the moving circle is one-third of the radius of the fixed circle : see Todhunter's Integral Calculus, Art. 112. When y = 0, we have either < = or < = -^ : at present we shall discuss the former case. 217. We suppose then that < = when y = 0, and we measure s from that point; thus ^ s = Q (1 - cos 3$). o Let <>! be the value of < when y = b ; then to determine c and <, we have the following equations : [a TT I y 2 dx = the given volume ; JXQ the latter equation may be written 7TC 3 I * sin 6 cos 8 $d$= the given volume. J o Effecting the integration and substituting for c, we find that the expression on the left-hand side is sn cos It is obvious that the value of this expression can be made as great as we please by taking small enough : but the value cannot be made as small as we please, for of course it is greater than and it may be shewn that this is always positive, and has its least value when $ = ^ , namely ^r . ^ J _') WITH GIVEN VOLUME. 199 Hence the solution we are considering becomes inapplicable when the given volume is less than a certain definite limit which may be easily assigned. For put the expression found above for the volume in terms of tan^; it will be found that the expression becomes , 3 /tan 3 j 3 ' il20~ ~20~ + 8t by differentiating with respect to fa we find that this expression constantly diminishes as <, increases from up to ; the least o 7T value is when fa - ; the value is then 218. Let us now consider the term of the second order in the variation of u where u denotes the integral in Art. 216. We observe that since the limit x is not fixed, we must attribute to it a change or variation dx ', in consequence of this there occurs in Su the term -P c {^(4>) + 2 J X which we have not yet regarded. This term is of the first order in appearance; but as y, < (p), and $(p) all vanish when ? = X Q we may consider that it is not even of the second order in value. There is a relation between Sy and dx ; see Todhunter's Integral Calculus, Art. 359 : but it is here of no importance. The term of the second order in the integral is and by transforming this as in Art. 26 we obtain 200 SOLID OF MINIMUM RESISTANCE -y m for the term outside the integral sign involving (&/ ) 2 vanishes. But from (1) p 9 hence, by differentiation, that is 2X = yVOO; and so the term of the second order in Bu becomes or we may put it in the form IT ' P( *J*o Now " (p) is positive as long as p 2 is less than 3 ; and thus we see that if (3 4 sin 2 <) d(j> ; integrating between the limits and ^ we obtain Then for the continuous solution we substitute for c from the, equation b = c sin 3 1 cos <^ ; thus the resistance TT&' /3 . 2 , 4 . 4 ,\ = rr- -T sm 2 6. - ^ sm 4 t ; cos 2 ^ V4 o r V and j is to be found from the equation Trtfffa) = the given volume, where /(^J is put for sin <> cos 3 204 SOLID OF MINIMUM RESISTANCE that is, for For the discontinuous solution we put fa = ^ > an( ^ * obtain the whole resistance we add TT (b z y*) ; thus the whole resistance 77TC 2 . / 71 C 2 \ 137TC 2 where c is to be found from the equation 137TC 3 ., -J22Q = the g lven volume. 226. Let us take for an example the case in which the given volume is ^ . In this case in the continuous solution we put <^ = ~- ; and we o find the resistance to be -^r In the discontinuous solution we put 1920 5 this gives c 2 =& 2 x ^| lo and thus the resistance is we find this to be approximately 7rb* x '44012. Therefore the resistance is less for the discontinuous solution than for the continuous solution. When the given volume is -^- b 3 the so-called discontinuous solution coincides with the continuous : we have . = T , and the 4 . 77T&* resistance is -- WITH GIVEN VOLUME. 205 227. We shall now shew that both for the continuous and the discontinuous solution the resistance decreases as the given volume increases ; and that when both solutions are applicable the resist- ance is less for the discontinuous solution than for the other. Let F denote the volume ; let R denote the resistance for the continuous solution and S for the discontinuous solution corre- sponding to the same volume F. Then COS 2 (^ hence we obtain COS 2 17 sn Also F= 7T& 3 \ ^^ tan 8 & + ^ tan + ^ cot d>[ ; (l^U ZO o j hence we obtain dV TTb* ~TT Af\ 2 t TT (1 ~ 16 COS (&J. d^ 40 sin ^>j cos <, v Thus ^Pr = T sin 3 j cos Let BA represent the generating curve for the continuous solution, and DE for the discontinuous solution corresponding to the same volume. These curves then are similar; but it is obvious that DE is on the larger scale. Thus c is greater than 7, and so 7 is less than c. Therefore -^ is numerically less than -. Now we know that when V= ^ we have 8 less than R ; and when V-. 30 we have S=R. It follows that for all intermediate values of V we must have 8 less than R. For if R could be equal to 8 for any intermediate value of V, then as V increased R would decrease more rapidly than $; and thus R could not be equal to 8 when V became equal to ' . 228. It will be seen that we obtained in the preceding Article for -=, and - results of the same form; this at first sight WITH GIVEN VOLUME. 207 may appear strange : but the reason for it can be easily assigned. Suppose in the preceding diagram we were to draw just above DE the hypocycloid for the discontinuous solution corresponding to a slightly increased volume : let it be FG. Let S denote the resistance on ED and the corresponding part of CD produced ; and let V denote the volume. Then the prin- ciple on which the problem in the Calculus of Variations is solved is that of making $(S+ 4XF) = 0: see Art. 222. Thus 88= - 4XS F. This is true for all variations, and therefore for the variation by which we pass from DE to FCf. This result expressed in other notation becomes ^^ = 4X = . dV c Similarly we account for the result which is expressed by ^_ _ dV'' y' Q 229. As another numerical example suppose tan ^ = - . Then, 2i 113 having the meaning of Art, 225, we find that /(^) = OA- To determine c we have ~ = ? b* ; so that c = *> It will be found that for the continuous solution the resistance is 4597T& 2 1040 = 7r6 2 x -44134, and for the discontinuous solution the resistance is 320 208 SOLID OF MINIMUM KESISTANCE 230. Let us now briefly consider the other case which pre- sented itself in Art. 216, namely, that in which we suppose <= -^ 2t when y = 0. If we measure 5 from this point we have s = - cos 3. o As before we shall have c sin 3 fa cos fa. And fa must lie between =- and ^ in order that p may be of O '- invariable sign which we have assumed as a condition. As before we have 7T& 3 1 sin 6 J IT sin 9 cos 3 the given volume. To effect the integration conveniently in this case we put the integral in the form cos 3 (1 - cos 2 <) 3 (4 cos 3 - 1) sin WITH GIVEN VOLUME. 209 hence taking the integral between the limits we get cos 6 + cos 4 --cos 2 4 = the given volume. It is obvious that the value of the expression on the left-hand side can be made as small as we please by taking ^ near enough to ~ ; but the value of the expression cannot be made as great as we please, 6. being restricted to lie between ~ and ^ . In fact, O Ij denoting the expression by ?r6 3 jP((/) 1 ), we shall find that 1 and that ^F = u - VT M 6 * + 5 cot 4 + 10 cot 2 + 10) ; . ad> 40 sin o> *7T 7T so that .F() decreases continually as increases from - to . Thus the greatest admissible volume corresponds to l = - > and . Trb 3 x 217 1215 V3 ' 231. Since in this case p is infinite when y = 0, our investi- gation of the variation of the integral to be considered is not satisfactory, and it will be better to take y as the independent variable. The investigation will be given presently and will lead to the conclusion that we have now a maximum. But of course this result must be understood with due restriction. We must not suppose that we have thus a solid of greatest resistance ; for the greatest resistance is obviously when the solid is a cylinder, or, which amounts to the same thing, when the solid has the step- shaped boundarj^ indicated in Art. 205. The statement merely means that the resistance is a maximum with respect to any admissible variation. A greater resistance can be obtained imme- diately by replacing the boundary by a figure like that in 14 210 SOLID OF MINIMUM RESISTANCE Art. 205 ; this is however not an admissible variation, because although the value of Sy might be made everywhere infinitesimal, yet that of Bp could not. 232. Let us now give the formulae for solving the problem when y is taken for the independent variable. Let CT stand for -^ . Then the resistance = 2-rr I ^ ^ 2 , dy J 1 + VF rvi and the volume = TT I y^^ dy. > Thus we seek for a maximum or minimum of where X is a constant. The complete variation to the second order, supposing the upper limit changed from y, to y t + dy l is The first term is made to vanish in the usual way by putting equal to a constant, which constant must be zero. This of course leads to the same results as in Art. 216. The case which we proceeded to discuss in Art. 217 is best treated as it was there by taking x as the independent variable. The present investi- gation will be suitable instead of that which began in Art. 230, as it avoids the infinite quantity which there occurred. We see, however, that if we do not suppose dy^ = 0, we do not make the term of the first order in the variation vanish; but if we take dy t = the term of the first order does vanish, and the variation reduces to i ?J*r' dy, WITH GIVEN VOLUME. 211 which is negative, because CT* is less than - throughout the o integration. Hence we have a maximum for admissible variations. 233. There is a peculiarity in the investigation of the pre- ceding Article which requires notice ; a particular variation is inadmissible which might appear at first sight to be admissible. Let BA be a curve ; let BD be parallel to the axis of #, and suppose it indefinitely small : let DE be another curve. Then we could pass from such a curve as BA to such a boundary as BD and DE by means of infinitesimal changes Sx and &ET ; though we could not by infinitesimal changes in y and p : but nevertheless the variation is not one that our investigation of Art. 232 will allow. For we take TT I #V dy to denote the volume generated by the revolution of BA round the axis of x ; and then we should be taking TT I y* fa + &ST) dy for the volume generated by the revolution of BD and DE: but it is obvious that the last expression really represents the volume generated by the revolution of DE, and so our expression would omit altogether the volume generated by the revolution of BD. In order to allow for such a change of figure we ought to use for the general expression of the volume not 142 212 SOLID OF MINIMUM RESISTANCE but 2-7T I (a x)y dy. Then we proceed to find a maximum or a [ yi ( y \ minimum of / - ' ^ + Xoy ^xy } ay, where X is a constant. J V-L + > / This leads in the usual way to therefore -w+ (1 = a constant, and the constant must be zero. But now we have an integrated term, namely , . (1 + -57 ) This vanishes at the lower limit, for which y = ; but does not vanish at the upper limit. For such a variation as the diagram represents we have &c negative at B } and thus we have a positive term in the variation of the integral. Hence when we say that we have a maximum in Art. 232, we must remember that such a variation as that illustrated in the present Article is excluded : such a variation in fact would increase the supposed maximum. 234. In Art. 221 we proposed to return again to the dis- continuous solution in order to remove the arbitrary appearance which seemed to belong to it. Let BG and CA be the two components of the discontinuous solution. "When we take x for the independent variable as in Art. 216 this solution is not very clearly suggested ; the relation p = oo WITH GIVEN VOLUME. 213 which belongs to the rectilinear part BG does not satisfy the fundamental equation unless we also make X = 0, and this will not suit the curve part GA. Although the discontinuous solution does not appear to be suggested by the fundamental equation, yet we may be naturally led to investigate it by the consideration that the given base con- stitutes a boundary which is not to be transgressed by our gene- rating curve. We have already had examples of this character. Sometimes the boundary is suggested by the fundamental equation, and sometimes not : see Articles 68, 100, and 111. But, by making x the independent variable, and ascribing variations to y only, we have in fact put it out of our power to recognise the discontinuous solution. Let the dotted curve represent a curve lying close to the discontinuous solution ; then we cannot pass from one to the other by infinitesimal changes in y, and so the method we adopt is really unsuitable to the full dis- cussion of the problem. Accordingly to bring the discontinuous solution into notice, while retaining x as the independent variable in the investigation, we have to modify our expression for the resistance ; see Art. 222. Now turn to the solution of Art. 232, in which y is made the independent variable. Suppose, as there, that for the curve part we have (1 + '37 ,2\2 then the term of the first order in the variation vanishes so far as the part GA is concerned ; and for the part GB it reduces to that is, since ^ = 0, to 41 \y Sx dy. J o As Bx is necessarily negative, this is essentially positive, and so corresponds to a minimum. Hence we have simply another 214 SOLID OF MINIMUM RESISTANCE illustration and confirmation of the general theory laid down in Art. 17. 235. Suppose we add another condition to the problem stated in Art. 215 and restricted as in Art. 220 : let the height be given, that is, the distance of the vertex from the base. If the height of the continuous or discontinuous solution already found is less than the given height, the solution already found must still be adopted ; and to produce the required height O o a portion of the axis of x must be used. Thus in the diagram, if G is the given height, we must add the straight line OA to the curve part AB. It will be seen that y is a solution of the fundamental equation (1) of Art. 216. If the given height be less than that of the continuous or discontinuous solution already obtained, we may still use a portion of the axis of x to fulfil the prescribed conditions in the manner of Art. 134, when the cone points inwards. Or if we understand the condition in another way, we must take as part of the boundary a straight line at right angles to the axis of x and proceed in the manner of Art. 190. 236. We may here notice the more general problem in which it is required to make / y $ (p) dx a maximum or minimum while I ^ (y) d& is constant, supposing that i|r (y] vanishes with y, that the value of x at the upper limit of integration is fixed, and that y is to vanish at the lower limit of integration. WITH GIVEN VOLUME. 215 We have to find a maximum or minimum of XQ where X is a constant. Denote this by u ; change x into X Q + dx Q and vary y and p : then Xo+dx x = f J - J ?p) +x^ (y + %) -y* (p) - f J This is exact; now approximate to the second order: then the first of these integrals becomes in the usual way S4>'(p) all to be taken between the limits * and a. The second integral becomes - \ for we know that to the second order X (*) * = fo () & + 1 X 216 SOLID OF MINIMUM RESISTANCE To make the term under the integral sign of the first order vanish, we require in the usual way and this leads to U$ (P) + W (y} = yp'(p) + constant; this constant must be zero, since zero is to be a value of y and ^ (y) vanishes with y. The terms of the first order outside the integral sign obviously vanish, except y ^ + **' (p) ^ * + y*" w ( w dx J XQ - !<#> (P) % + yf(P) *P + X^r' (y) tyl cte. - IP* (p) + yy"'(p) + ^f ' ()} Now the following relation, true to the second order, exists between the differential and the variations See Todlmnter's History of the Calculus of Variations, page 330, observing that the ty (x) there is now simply x. Bat we only require now the relation &/ = (pdx\ which is true to the first order. Thus observing that ?/ =0, we find that the part of the term of the second order in the variation which is outside the integral sign becomes WITH GIVEN VOLUME. 217 The part of the term of the second order in Su which is under the integral sign may be transformed, as in Art. 26, to [$'(p) This may be modified in form by the aid of the fallowing relation, ^C + ^k > */;,K O 237. As an example of the preceding general investigation^/ take (p)=p* and ^(y)=y z \ so that \yp z dx is to be a mini- mum, while I y z dx has a constant value. And let us suppose the value of y given at the upper limit. We have y$ (p) + Xt/r (y) yp<$ (p) + constant (1) ; that is, since the constant must be zero, yp* + \y z = 2i/p 2 ; therefore P*ty* (2). For X put T 2 ; thus from (2) we get _ yy ' therefore y = Be^ x where B is a constant. The constants B and 7 will have to be found from the known value of y at the upper limit, and the known value of I y z dx. In this example, supposing 7 to be positive, we have y zero when x = o ; so that x oo . 218 SOLID OF MINIMUM RESISTANCE The conditions for a minimum may be considered to be satisfied ; as we see by taking the term of the second order in $u in the last form of the preceding Article. But we do not thus get the least value of \yp z dx\ for as in Art. 219 we can get as small a value as we please of this integral by a zigzag boundary: this boundary is suggested by the fact that the fundamental equation (1) is satisfied by a constant value of y. Suppose that we impose another condition in the manner of Art. 235 ; let it be given that the curve is not to extend on the negative side of the axis of y. Then we must seek for a solution by combining a portion of the axis of y with the curve given by (2). Thus we obtain a figure composed of pieces like the OA and AP of the diagram of Art. 201. The solution must be taken as in former examples to be a limit towards which we approach by drawing curves close to the discontinuous boundary. The curves must of course be supposed so taken as to make I yp* dx finite in the neighbourhood of the origin ; for example, this will be secured if near the origin y varies as \/a?. 238. The problem of Art. 236 may also be treated by taking dx y as the independent variable. Put CT for -y- , and let (p) dx transform into /(CT) dy ; then we have to find the maximum or minimum of This leads in the usual way to Sf + >*(y)-0 ..................... (1); and if y^ is susceptible of an increment we must also have ...................... (2)- WITH GIVEN VOLUME. 219 Then we have left a variation of the second order consisting of together with the part of the second order in r +dy Jyi the latter part is + 1 {/ H + y/' W + vf (y) + which by means of (1) reduces to This may be modified in form by the aid of (2) CHAPTER XL JAMES BERNOULLI'S PROBLEM. 239. A and B are fixed points ; a curve of given length is to be drawn from A to B, having the following property: at any point S of the curve draw SN perpendicular to the fixed straight line OF, and take the ordinate NP equal to the arc AS; then the curve traced out by P is to enclose a maximum or minimum area. The problem is a particular case of one which was given by James Bernoulli: see Todhunter's History of the Calculus of Variations, page 453. JAMES BEKNOULLl'S PROBLEM. 221 240. It will be apparent on a little reflection that there must be both a greatest and a least value of the area bounded by OPQ. The fixed straight line OF will be taken for the axis of x, and the axis of y will be taken to pass through A. We may remark that by the nature of the curve OPQ, the tangent at any point P cannot be inclined to the axis of x at an angle less than 45. We assume that the curve A SB is to be comprised between the ordinates at A and B. Let OA = h, OF=a, FB = k. Let x and y be the co-ordinates of S, and let s denote the length of the arc AS: then ra and we require that sdx shall be a maximum or minimum, Jo ra while I J(l+p*)dx inconstant. J o ra Let w= I {s + \ V(l + P*)} dx where X is a constant. Change p to p + Spi then to the second order f f, \p%> MSp) 1 ) , And ISs Jo? = ccSs - \x ^ cZ^, so that = 1 (a x] J o dx. 222 JAMES BERNOULLI'S PROBLEM. Thus, finally, From the value of $u we see that our problem coincides with the following: find a curve of given length between the fixed points A and B for which I (a x) ds is a maximum or a minimum. Hence, we require in effect a curve of given length which shall have its centre of gravity at a maximum or minimum distance from the straight line x a. The curve is well known to be a catenary. The term in $u which is of the first order must now be transformed in the usual way : we have d When this is taken between limits the part outside the in- tegral sign vanishes; to make the other part vanish we must put (A, 4- a x) p a sayc ............ ^ ; therefore ^-^- ?> " (2). pc Thus SM reduces to the term of the second order ' x+ (i a +j>* P *" 241. The curve determined by (2) is a catenary having its directrix parallel to the axis of y ; and c is numerically equal to the parameter, which, in the language of Statics, measures the tension at the lowest point. By putting x = a in (2) we see that X is numerically equal to the perpendicular distance of B from the directrix of the catenary. But we shall have to pay careful attention to the signs of c and X. JAMES BERNOULLI'S PEOBLEM. 223 From (1) we have therefore thus c is of the same sign as ^ , so that c is positive or negative dx according as the required curve is convex or concave to the axis of x. 242. We will now assume than h is less than k ; from our discussion of this case it will be easy to see how to proceed when h is not less than &. If then the curve A SB is concave to the axis of x, we have c negative ; and then X + a x is negative by (1) ; thus X is negative. And since X + a x is negative, Su is a negative quantity of the second order, and we have a maximum. If the curve A SB is convex to the axis of x we have c positive ; and then X + a x is positive by (1), and so \ is positive. And since A + a x is positive, $u is a positive quantity of the second order, and we have a minimum. 243. Now let us consider how far these solutions are really admissible. Take for instance the maximum. Begin with a given length very little greater than the straight line which would join A to B\ it is obvious from statical considerations that the required catenary exists. Suppose the given length gradually increased ; then it is obvious in the same way that the required catenary always exists until we arrive at the case in which the catenary touches OA at A. If the given length be still further increased, the catenary would cut the axis of y above A, and the solution is no longer tenable. 244. Now, as by supposition we are confined by the axis of y, we are, as in former problems, led to enquire whether the solution will not be composed in some cases of part of the axis of y, together with a curve ; see Arts. 221 and 234. 224 JAMES BERNOULLI'S PROBLEM. Suppose then, if possible, that the required line consists of a straight line of the length y Q h measured from A along the positive direction of the axis of y, and a curve proceeding from the point in the axis of y which has y Q for ordinate to B. Thus we now ask for a maximum of ra (y Q -h + s)dx, J rx where s = V(l + p 2 ) dx ; Jo and the given length must be equal to o Therefore we have now to find, a maximum of that is, of The only difference between the present form of the problem and that of Art. 240 arises from the presence now of terms in the variation outside the integral sign. We have (X + a) Sy from (X + a) (y Q fi) ; and from the integral itself we obtain cSy Q : thus, on the whole, we have (X + a c) &y . Hence, as this must vanish, the following relation must hold, X + a_c = (4). Now, the curve being supposed concave to the axis of x, we know that X is negative, and X + a is numerically equal to the distance of A from the directrix of the catenary : hence it follows from (4) that the catenary must touch the axis of y at the point corresponding to y Q . Thus the conditions required for a maximum are all satisfied in this solution. JAMES BERNOULLI'S PROBLEM. 225 This solution holds only so long as y Q is not greater than k, the limiting case being that in which y = k, and the catenary degenerates into a straight line : the given length is then equal to k h + a. 245. It should be remarked that the solution of the preceding Article is suggested by the general equation (1) of Art. 240. For if we suppose x = so that p is infinite and put X + a c, that equation is satisfied. And the relation A, + a = c holds, as we have seen, for the curve part of the solution also. Thus the discontinuous solution arises in fact from combining two solutions which are both involved in the ordinary general result of the Calculus of Variations. In Art. 234 we had to account for the fact that the discon- tinuous solution did not appear to be very obviously contained in the general result ; whereas in the present problem the discon- tinuous solution is so contained. The difference perhaps depends on the fact that here, as we see in Art. 240, we do not imperatively require that y should be infinitesimal in our investigations, for y does not explicitly occur under the integral sign: we merely change p into p + $p, and our process requires that &p should be indefinitely small in comparison with p, so as to allow us to expand Vl + (p + 8p) 2 suitably. 246. We may observe that for a given length of string there is only one solution which furnishes a maximum ; namely, either the continuous solution of Art. 242, or the discontinuous solution of Art. 244, according as the given length is less or greater than a certain value. This will be sufficiently obvious from statical considerations. Suppose a fixed point A on a smooth horizontal table, and a fixed point B above it ; let a heavy uniform string have its ends fastened at A and J5; and suppose the length of this string to be not less than that of the straight line which joins A to B, and not greater than the k h -f a of Art. 244. Then we may admit that this string will be in equilibrium in one position and only in one : if the length is below a certain value no part of the string will be in contact with the table, and if the length is 15. 226 JAMES BERNOULLI'S PROBLEM. above this value, part of the string will be in contact with the table. If I denote the length of the string when the string touches the table at A, we find from the nature of the catenary that where 7 is put for k 7i. Of course these statements may be demonstrated by analysis without having recourse to statical considerations. When the length of a string and the positions of its extreme points are given, we can form equations for determining a catenary with its base horizontal fulfilling the given conditions : we find that there is only one value of the parameter c, and the greater the given length is the less is this parameter. For let I denote the given length of a string; let h denote the horizontal distance of the fixed points ; and let b denote the difference of the distances of the fixed points from a given hori- zontal line. Then it is shewn in works on Statics that the para- meter c of the required catenary is determined by the equation Expand the right-hand member in powers of c ; thus we ob- tain 1 A* 1 h 4 The expression on the right-hand side decreases continually as c increases ; and thus we see that the equation will give only one value of c 2 corresponding to an assigned value of /, and the greater I is the smaller c 2 is. Hence two catenaries with parallel bases cannot intersect in more than two points ; for if they could each would have a less parameter than the other, which is absurd. Hence as the given length is gradually increased from the least admissible value, a series of catenaries is obtained each lying entirely outside the JAMES BERNOULLI'S PROBLEM. 227 preceding, until we arrive at the length denoted by I in the former part of this Article. Then after passing this length we can obtain another series of lines each consisting of a straight line and a catenary. Now consider two members of this series. If the catenary corre- sponding to that which has the longer straight line be continued upwards from its lowest point, it will obviously cut the other catenary ; and as the two catenaries have also the fixed point in common they cannot cut in any third point. Hence of the two members of the series that which has the longer straight line is the longer; for it is outside the other except where the two coincide. Hence it follows that corresponding to a given length there is only one member of the series. Thus as there is only one solution for a given length of string which furnishes a maxi- mum, and as we are sure there must be a greatest value, we may infer that the maximum value is the greatest value. 247. Let us now return to that point of the investigation which was reached at the end of Art. 244 ; and suppose that the given length is greater than k h + a. We shall find that there is now a maximum when we take a straight line of length y Q h along the axis of y from A, and a catenary convex to the axis of x touching the axis of y at the point corresponding to y , and passing through B. The investigation is similar to that of Art. 244 ; now we have c positive by equation (3), and as p is negative we see that \ + a x is negative by equation (1). As in Art. 244 we arrive at the condition X + a c =$, which makes the catenary touch the axis of y at the point corresponding to y Q . But although we thus obtain a maximum, we do not obtain the greatest value of the area ; for there is another solution which is now admissible. 248. We see in fact that p =0 is a solution of (1) provided of course that c = ; and we will now interpret this in combination with p = oo , which is also a solution of (1) supposing that X + a x = since c = 0. 152 228 JAMES BERNOULLIS PROBLEM. Draw BG parallel to the axis of x ; take a point D on the axis of y such that AD + DC + CB = the given length ; then we may suppose that our required curve is made up of AD, DC and CB. Corresponding to the curve OPQ of Art. 239 we have now the straight line EPQ where OJR = This solution might be treated in the manner of Art. 244. We now ask for a maximum of ra (2/o - ^ + 2/o - & + 5 ) d> J o rx where s = I VO- +.P 2 ) d > J o and the given length must be equal to y, - * +2/0- & + f a V(i +/) fo. ./ o Hence proceeding as before we arrive at the equation (X + a x) p ^ - if* = a constant c, and we have as the term in the variation outside the integral sign JAMES BERNOULLI'S PROBLEM. 229 Hence we obtain a solution by the suppositions p = 0, c = 0, a + X = 0. 249. With regard to this solution I remark I. It is certain that we obtain the greatest possible area in this way. For the ordinate of Q the extreme point of the derived line is of course equal to the whole given length ; and the derived line becomes in this case a straight line inclined at an angle of 45 to the axis, and so the ordinates of this derived curve starting from Q diminish more slowly than for any other possible form of the primary curve : see Art. 240. II. This solution, like the solutions for the cases already considered, is fairly deduced by the Calculus of Variations. III. Should any person object that the solution does not strictly apply to the problem, for we were required to draw a curve from A to B, the answer must be similar to remarks already made. The proposed solution must be regarded as a limit. We may conceive a curve drawn from A to B, first running upwards close to the axis of y, then turning sharply and descending close to the ascending part until it arrives at about the level of B, then turning off towards B in a direction nearly parallel to the axis of x. Such a curve will give us for the area of the derived curve a result falling only infinitesimally short of what we have shewn to be the greatest possible value. 250. We will now briefly state the result with respect to a minimum. If the given length be very little greater than the straight line which would join A to B, we are certain that a catenary can be drawn from A to B convex to the axis of x. This will give a minimum. The solution will hold up to the limiting case in which the catenary touches the ordinate of B at B. If the given length be greater than that which corresponds to this limiting case the required line is made up of a portion k y^ of the ordinate at B measured from B towards the axis of a?, and a 230 JAMES BERNOULLI'S PROBLEM. catenary which touches the ordinate of B at the point correspond- ing to y v This will give a minimum. The solution will hold as long as y^ is greater than h. In the cases hitherto considered there is only one solution for a given length ; and the solution is not only a minimum, but corresponds to the least area of the derived curve. If the given length is greater than k h -f a we have a choice of two solutions. One consists of a portion k y v of the ordinate of B and a catenary concave to the axis of x, and touching the ordinate of B at the point corresponding to y l which is less than: h. This solution gives a minimum. The other solution consists of the straight line y = h together with such portions of the ordinate at B as may be required in addition to produce the given length. This solution corresponds to the least area of the derived curve, as we see by considerations like those already employed, 251. It is easy to give other examples like that of Art. 248. Return to the diagram of Art. 239, and suppose we require not that the area OPQF shall be a maximum or a minimum, but that the volume generated by the revolution of this area round Ox shall be a maximum or a minimum. Retaining the same notation as before, since the volume will- be TT Is 2 dx, we have to find a maximum or minimum of Denote this by u ; then to the second order And \ss dx = vSs Iv-j- rx where v stands for I sdx\ so that Jo 2(1 JAMES BERNOULLI'S PROBLEM. 231 r = sda;. JQ where F= sda;. Hence Hence we obtain in the usual way fa (mV f a and then Bu reduces to I (X + V- v) ^ P) s dx + (Ss)*dx. Jo (1 +'y) J& From (5) we have X+ F-c = dx P differentiate with respect to x ; thus Multiply by ^- and integrate ; thus constant The equations (5) and (6) are not simple enough to furnish us with the relation between x and y\ but for our purpose the most interesting point is that if the given length be large enough, we can solve (5) by the combination of p = with p = oo in the manner of Art. 248. We first take x = which gives v = and p = oo ; also we make X + F= so that c = 0; then we take p = 0. Thus 232 JAMES BERNOULLI'S PROBLEM. &u becomes I 2 % which is necessarily negative ; for ra I ($s)*dx vanishes since p = 0: so that we have a maximum. J Q And by the reasoning of Art. 249 we are sure that this is not only a maximum, but that it is the greatest possible result. 252. Let us propose a problem in polar co-ordinates like that of Art. 239. An arc ASS of given length is to join the fixed points A and B\ on 08 take OP equal to the square root of AS: required the curve A SB so that the polar area generated by OP may be a maximum or a minimum. The area generated by OP- -^\(OPfd6, and therefore it varies as [sdO where A8 = s. Let u = ffc + \V(r 2 +/)} dO, where X is a constant, OS=r, and p stands for -^; also s= I */(v*+p*) a ^- au Jo Proceed as in Arts. 239 and 251 ; thus we shall find that to the first order JAMES BERNOULLI'S PROBLEM. 233 This must be made to vanish in the ordinary way by the relation \+a-0r d _ Then the term in Su of the second order is (X + a- thus X 4- a 6 must be always negative for a maximum and always positive for a minimum. Although we cannot deduce from (7) the explicit relation between r and 6, yet we shall see that if the given length be large enough we must have discontinuity of the kind exemplified in Arts. 244, 247 or 248. Equation (7) when developed becomes hence we can infer that r does not admit of a maximum or a dr -^ dr minimum at any point between A and B. For if -^ could change Cdr\ 2 d 2 r ~ia ) ~~ r ~jni must change sign at the same point ; that au/ do is, there would be a change as to concavity and convexity at the point where r is a maximum or minimum : and this is impossible. Hence we conclude that equation (7) will not supply curves of unlimited length between A and B ; so that when the given length is large enough we must seek for a discontinuous solution. If we proceed as in Art. 244 we shall have outside the integral sign in $u the term (x + a_0) H- a - // o . ^H or 234 JAMES BERNOULLI'S PROBLEM. To make this vanish we must either have X + a = or jp infinite. If we take p 9 infinite we have a curve touching the initial radius OA at the point where the curve leaves the radius; and thus the solution resembles those in Arts. 244 and 247. If we take \ -f a = we see from (8) that p = ; in this case our solution resembles that of Art. 248. If we start as in Art. 248 we have for the term in $u outside the integral sign and to make this vanish we must have X 4- a = 0. 253. We might approximate to the solution of equation (8) in particular cases. For example if we take X -f a = 0, then we shall find that for small values of 6 we have approximately where b is the value of r corresponding to = 0. 254. The term of the second order in Su of Art. 252 will vanish if p$r r$p is always zero ; and thus it might seem as if we should not be sure of having obtained a maximum or a minimum. But this relation leads to Sr = Cr where C is a constant ; and it will be found that this is inconsistent with our condition that the length of the curve is given. 255. Let p stand for ^- and q for -^- : required a maximum or minimum of while I" I V(l +p* + f) dx dy Jo JQ is constant, and J JAMES BERNOULLI'S PROBLEM. 235 This is an obvious extension of the problem of Art. 239 to space of three dimensions. By the usual theory we must seek for a maximum or mini- mum of r r [s+ \ va +P* + ' (s) dx : then JO = ^ (aj) s -^(x)~ dx, so that Hence 8u = J" {* (a) - ^ (*) + + 1 /* [f () -*() + 238 JAMES BERNOULLI'S PROBLEM. To make the term in Su which is of the first order vanish, we put As in the special case of (s) = s, which we have already discussed, we see that there may be particular admissible solutions of (10) given by p = and p = oc , as well as the general solution. We may put (10) in the form (a) -() + X = ............... (11); therefore, by differentiation, / / \ _ __ c dp ,, ,,,. ds c dp so that } ( 2 Jo 1 (l+/) f P W/ taking the general solution (11), this becomes If - 8 is of the same sign as-#"(s) this term will require P no transformation ; if - 3 is not of the same sign as 0" (s) the following transformation may be useful: JAMES BERNOULLI'S PROBLEM. 239 and as &s at the limits, we get / c fdSs\* , f% d (c dSs\ - &e = os -j- - 3 -j- jQ\dxJ Jo dx\p* dxj \p 5 \dxj Jo dx\/ so that we have 2) * ^ t /"//> \ /Y\^ J o ( ax \p 257. From equation (12) we obtain -- = therefore dx c, and -y- = From these last two equations we must find x and ^ in terms of s by integration. Thus from the last we obtain x + c 2 as a function of s, c and c,, where c 2 is another constant, so that we may say we have s=f(x + c 2 , c, cj ..................... (13). And in like manner we should have c z , c, cj ............ . --------- (14), where c 3 is another constant. We have thus four constants c, c lt C 2 , c g ; these must be de- termined from the four conditions x = Q when s = 0, x = a when 8 has its given length, y=h when s = 0, y k when s has its given length. The equation to the curve would be fouad by eliminating s between (13) and (14). Then from (11) vwe could find \ by ascribing any value we please to #; for ; instance, if we put x = a, we see that \ is the value of c - - when x = , and this of course is known since the equation to the curve is known and c is known. 240 JAMES BERNOULLI'S PROBLEM. 258. The form in which we have left $u at the end of Art. 256 suggests to us to ask if we can apply Jacobi's method here. It is obvious that the occurrence of s in the problem of Art. 239 renders the problem different from those which merely involve y and its differential coefficients for which Jacobi's method is specially constructed. By (13) we have c, cj. yy O fJ O Now let z stand for ^ or for -= ; then according to the principles of Jacobi's method, we see that if z be put for Ss in fr d f c dSs the equation is satisfied. At first sight we might also suppose that if the value -j- were put for Ss, the equation (15) would be satisfied ; so that apparently three forms could be found for 8s ; but it is obvious that there cannot be more than two different forms, inasmuch as a linear differential equation of the second order can only have two particular forms of solution. And in fact, since the quantity c itself occurs in (15) it will be found on ds examination that we are not justified in concluding that -j- will be a solution of (15). It is easy to verify that -~ is a particular solution; for by cCc 2 (13) we have df == df_cte dc z dx ~ dx ' ds We have to shew then that (15) is satisfied when -j- is put for Ss. JAMES BERNOULLI'S PROBLEM. 241 rNow 13! pV(i+^ 2 ) ^ a ;' *" M ^ 5 J f c ^l there! 01 e * (s] dx dx]f */(!+# yacj. 1 ( jds\ tlmt is <>" M ~ = die dx\ This gives the required verification. Since then we have a particular solution of (15) we can easily ds find the general solution. Denote it by t-r- : substitute in (15), and use (] 6), thus we obtain (' (q) should vanish at both limits : this leads to (7,=0 and <7 2 = 0; therefore <' (g) = ; that is, in this case Hence q = + - ; and therefore we have where /3 = - , and \ and jj, are constants. a Then, in order that the curve may pass through the given points, we must have /-t = 0, and \ must be found from yi = -Y The term of the second order in Su is and this is necessarily positive ; so that we have a minimum. In fact we have here not only a minimum, but we have the least possible value of the proposed integral. For and this has its least possible value when q? = + - ; and so the a integral has then its least possible value. But it must be observed that we can give discontinuity to our solution; and make it consist of a broken or zigzag path. Instead of going along one parabola from the first fixed point to the second, we can take arcs of different parabolas, all being MULTIPLE SOLUTIONS. 247 given by an equation of the form where /3 may be at our pleasure either - or -- , and X and fju are constants. It is obvious in fact that we get the same value for the integral as before. This amount of discontinuity might indeed have been antici- pated; for since u involves only q z , we might have expected that so long as the value of q was numerically unchanged the integral would remain unchanged. 264. Suppose now that we modify the problem by making the curve touch given straight lines at both the fixed points. In this case Sp is zero at the fixed points; thus we have no longer (7 t = and (7 2 = 0. Hence a solution is to be obtained from the two constants which are here expressed, and the two more which will enter in the value of y in terms of x, must then be determined by the conditions that the curve is to pass through two fixed points, and touch fixed straight lines at those points. This solution will give us a minimum, but it will not give us the least value of the proposed integral. For we can still find a discontinuous solution; we may suppose it composed of the two parabolas fi and where /3 = - , and we may use either value in either equation. 248 MULTIPLE SOLUTIONS. That is, we start from the origin on the first parabola, and continue on it up to the point where the two parabolas meet ; and then we proceed along the second parabola to the second fixed point. The constant 7 must be determined so as to make the first parabola touch the given straight line at the origin ; and the constant X must be determined so as to make the second parabola touch the other given straight line at the second fixed point. All the conditions for a minimum are satisfied by this solution ; and in fact we see, as before, that it gives us the least possible value of the proposed integral. 265. It is obvious that a result of a similar kind to that in- the preceding Article will hold whenever we seek the maximum or minimum value of an integral which involves nothing except one differential coefficient. For instance, let r stand for -yj^ ; required a maximum or minimum of /$ (r) dx between fixed limits. Let u denote the integral ; then to the second order Now we do not assert that we must have ' (r) = ; because it is not obviously certain that Sr can have either sign consistently with such conditions as may be imposed at the limits; but we can always try if (/>' (r) = will give a solution. If ' (r) breaks up into factors, we can combine two factors ; say r = a and r = /3 are thus deduced. Then from each of these we can get a relation between x and y and three arbitrary constants ; thus on the whole we have six arbitrary constants, which is the number we should get from the ordinary continuous solution, namely, from __ (' (r) = 0. Thus we can in general make the discontinuous solutions satisfy as many conditions as the ordinary continuous solution. Compare Art. 262. The solution gives a maximum if " (r) be negative, and a minimum if " (r) be positive. 266. There is another way in which it is conceivable that discontinuity might occur in problems of the Calculus of Varia- MULTIPLE SOLUTIONS. 249 tions. Take the general equation M = of Art. 261, and try if we can employ two solutions of it, one with one set of arbitrary constants, and the other with another set. Of course the equa- tion M = of Art. 261 will be satisfied whatever be the values of the arbitrary constants. But when we consider the term L, we shall find that it will in general be impossible to satisfy all the conditions relating to this term with different sets of arbitrary constants. 267. The remark in the preceding Article, however, must not lead us to suppose that we can never make use of a com- bination of two solutions of the equation M 0, which differ in the constants involved : the next Chapter will furnish an illustration of the possibility of such a combination. In Chapter VII. we have also Examples of such combination; take, for in- stance, that in Art. 143. And a very instructive case, though of somewhat different kind, occurs in Art. 41 : here, in fact, the equation M = is satisfied by r = ^ ~\ > where /3 is a con- cos (v p) *TT stant which is zero for one part of the solution, and - for another JU part; but instead of shewing that the terms arising from L vanish, we shew that they are essentially negative. See also Art. 64. CHAPTER XIII. AREA BETWEEN A CURVE AND ITS EVOLUTE. 268. KEQUIRED a curve connecting two fixed points such that the area between the curve, its evolute, and the radii of curvature at its extremities may be a minimum. This is a well-known problem, which has however hitherto been very imperfectly discussed. We will suppose the curve to be concave to the axis of x, so that q is negative. Let -f&V I -q the integral being supposed to be taken between fixed limits. Then to the second order n J\ ,7 Bp i - - cq> dx The term of the first order in SM becomes by the usual transformation AREA BETWEEN A CURVE AND ITS EVOLUTE. 251 where M stands for 4 ,- ^1^ + -f- 2 ; a nd the whole dx q dx 2 be the angle which the tangent to the curve makes with the axis of x ; and assume G^ = k sin /3, (7 2 = k cos ft ; thus k p = 2 cos (

" (p) ?! dx, 264 MISCELLANEOUS OBSERVATIONS. so that the term of the second order in Su becomes 2 x (y) *' (P) + \ /()' % (y) <*>" (P) d ' fx" (y) H> (p) - pf (p)] - x (y) f (P) ?! Sometimes we may be able to determine the sign of the term of the second order in Bu from the expression just obtained. If the constant in (1) is zero we must have p constant, whether we take % (y) = 0, or (p) p$ (p) = ; so that q = 0, and this effects a great simplification in the term of the second order. If we take ^> (p) p therefore dy _ df dF / s Again from (2) and (3) dy df dp dc 2 dp dc z * /\^. *> ^ i S3^> ^ K% % 1- %i s 0. ~ dp dc z i ; < V/ -/. x> therefore dy -*. = - v. Sm ==/ f- Now -f- is to be obtained from (2) supposing^? constant; or we Cj may if we please obtain it from (1) which is equivalent to (2). Thus X and therefore Thus the quantity which was denoted by z in the account of Jacobi's method in Art. 24 becomes A j_x(f fl ^, u^rs) r *j where jB t and jB 2 are constants. This is as far as we can carry the general process, because we 777T cannot express ^- while % (y) remains quite general. aCj As an example suppose % (y) y" ; therefore, by (1), 3/ 266 MISCELLANEOUS OBSERVATIONS. ^ Thus we get x = c l n O(p) +c 2 , where (p) denotes some function of p which does not contain c l explicitly. Here then T> and z becomes 1 [y p (x c 2 )} B 2 p. nc \ Hence the same interpretation with respect to the tangents at the extreme points of the curve which we may suppose to be required holds as in the case of n 1 : see Art. 29. And the reason is obvious ; for by changing the constant we have in this case ;?/ = CT/T (p), where c is a constant, and ^ (p) is some function of p. 283. An important remark must be made with respect to relative maxima and minima which, so far as I know, is not to be found in treatises on the subject. Suppose we require that I udx shall be a maximum while I vdx is constant : then according to the usual theory we seek the maximum of I (u + av) dx where a is a constant. Now when we come to the term of the second order in S I (u + av) dx, it is quite conceivable that we may find it is not certainly negative ; and thus we infer correctly that I (u + av) dx is not a maximum. But still it may be quite possible that I udx may be a maximum : for the variations are really limited by the condition that I vdx is constant, and to this condition we pay no regard when we merely consider the term of the second order in I (u 4- av) dx. This remark is necessary in order to anticipate an objection which might be brought against some of our results. MISCELLANEOUS OBSERVATIONS. 267 For instance in Art. 96 we proposed a solution for a certain problem. Let AEDFB be the boundary of the solution there proposed. Suppose that the dotted boundary is obtained by giving an in- finitesimal increment to the constant e of the investigation. Then there would not be a minimum of I (%y J\ + p 2 j dx, because it is possible to draw between H and K a curve infinitesimally close to HDK satisfying the same differential equation. See the reasoning in Case II. of Art. 24. Nevertheless this does not shew that the surface is not a minimum for a given volume : it is obvious that the curve here obtained by variation does not satisfy the condition of generating the same volume as the original curve. The volume is in fact increased by taking the dotted curve between H and K instead of the other curve. I use the preceding only for illustration, so that it is not absolutely necessary for me to shew that such a line as the dotted line can be drawn. Nevertheless I think the following considera- tions will shew that such a line can be drawn. In the diagram of Art. 96 suppose a to remain constant, and e to receive a small increment. Then AE and D C increase, and AB decreases. Also the increase of AE bears a finite ratio to the decrease of AB\ this shews that the varied place of E is within AEDFB, as I have drawn it. Then the dotted line having crossed the other keeps above it up to D ; this we see by the argument of Art. 97. Then by symmetry the dotted line crosses the other again at a point K which corresponds to H. 268 MISCELLANEOUS OBSERVATIONS. 284. The Calculus of Variations is much stronger in its nega- tive results than in its positive results, that is to say, we learn from it rather when a given expression has not a maximum or a minimum value than when it has. Suppose that we have to find the maximum or minimum of vdx ; denote this by u : then we obtain in the usual way Bu = L then we say that there can be no maximum or minimum, if By be unrestricted, except when M ; for if M is not = we can make Bu positive or negative at our pleasure by taking By suitably. But of course this does not ensure a maximum or a minimum without examining the term of the second order in Bu. There is, strictly speaking, only one case in which a perfectly definite result can be obtained, namely, when we know beforehand that there must be a maximum or that there must be a minimum. Take the case of the brachistochrone between fixed points ; then the argument is as follows : we feel certain that there must be a line or lines of descent such that no other line of descent can be fallen through in less time ; but if M is not zero the time of descent can be made less ; therefore no curve can be the brachistochrone except a curve which satisfies the equation M = and passes through the fixed points ; thus we feel certain that a curve must exist satisfying these conditions, and that it is the curve we require. When this argument is put briefly it is often put inaccurately thus : we are sure that there can be no maximum in this case, and therefore the result must give a minimum. This is inaccurate, because we are not sure beforehand that there is no maximum in the technical sense of the word maximum ; we see that the time of descent can be made as great as we please by suitably adjusting the line of descent, but this does not justify us in asserting that there can be no maximum. But further ; suppose that we examine the term of the second order in a variation. If we find, for instance, that this term is MISCELLANEOUS OBSERVATIONS. 269 essentially positive, we can safely affirm that the relation which makes the term of the first order in the variation vanish does not give a maximum. But if we assert that the relation does give a minimum, we must bear in mind that this means a minimum with respect to admissible variations. Take for example the brachistochrone between fixed points. Suppose we draw close to the cycloid, which we obtain by making the term of the first order in the variation vanish, a line in the form of a series of indefi- nitely small steps, as in the diagram of Art, 205. Then the fact that the term of the second order in the variation is positive does not shew that the time down the cycloid is less than the time down the discontinuous figure, for our investigation is not appli- cable to such a variation as would be required in passing from the cycloid to the discontinuous figure: in such a passage Sp would not be always indefinitely small. Of course it might be possible to give some special investigation for such a case, but certainly the case is not included in the ordinary methods of the Calculus of Variations. When we assert then that a certain cycloid is the curve of quickest descent between two given points, the statement depends mainly on the fact that we feel certain beforehand that there is some curve which has the required property. Similar remarks apply to other problems; so that we can- not by the aid of the Calculus of Variations assert that we have the least or the greatest value of a proposed integral unless we are certain beforehand that such a least or such a greatest value necessarily exists. 285. There is still another consideration. Suppose that we are examining a certain curve to see if it possesses a prescribed maximum property. It may happen that at a certain point of the curve p is infinite ; the obstacle that thus arises in the use of the ordinary formula of the Calculus of Variations may prevent us from drawing the positive conclusion that there is a maximum : but such an obstacle may not prevent us from safely affirming that there is not a maximum. For we may of course apply the ordinary formulae to such parts of the curve as have p finite ; and 270 MISCELLANEOUS OBSERVATIONS. if the fundamental equation M = does not hold throughout such parts we are sure there is not a maximum. But if this equation does hold for the whole curve, and if the term of the second order in the variation is essentially negative, we cannot rely on our investigation so far as to assert that there is a maximum on account of the occurrence of the infinite value of p. Similar remarks apply if any of the other quantities which occur become negative. Take for example the brachistochrone between fixed points. If we make x the independent variable, as being measured horizon- tally, we have for the term of the second order in the variation the value given in Art. 281. But this is not trustworthy, for we have to vary - and p which are both infinite at the starting point. if If we make y the independent variable we have for the term of the second order the other value given in Art. 281 ; and this may be accepted without hesitation so long as OT is not infinite, that is, so long as we do not have to pass through the vertex of the cycloid in order to reach the second given point. 286. I have often spoken of the results which have been obtained as maxima or minima with respect to admissible varia- tions. I will give another problem to illustrate this point. A particle is to descend from one fixed point to another in a vertical plane, constrained by a smooth curve which is convex downwards : required the curve so that the integral I Pdt taken during the time of motion may be a minimum, where P denotes the pressure on the curve at the time t. Thus we may say that we require the whole pressure to be a minimum. Take the highest point as the origin, and the axis of x vertically downwards ; let v denote the velocity, p the radius of curvature, at the point (x, y] ; and let s denote the arc described up to this point. n j* // 77 Then P= + a --> -, and v = MISCELLANEOUS OBSERVATIONS. 271 Thus the integral = ((- + g &) dt J \p as J vds _ " ~- + d Put u for I \ .. + *!-) dx i then we require the minimum JV1+.P V/ of u. By the usual theory we must have dx ty* (1 4 p*)*) dx 2 1 + ,, f d 2 >Jx 4 as particular solutions. And the least value of the integral will be obtained by taking the portion of the axis of x from x to x = a, and then a portion of the straight line x a up to the lower given point. That is, the required line is made up of two straight lines at right angles to each other. MISCELLANEOUS OBSERVATIONS. 273 That this discontinuous solution is really a minimum may be shewn thus : There must be some line which gives the least value to the whole pressure ; this line must be a solution of (1) ; and the only solutions of (1) are the general solution given by (2), and the particular solutions p = and p oo . It is obvious that the general solution (2) produces a curve which is always convex doAvn wards ; and the discontinuous solution gives a less result than any curve which is convex downwards for two reasons : first, the element - is zero ; and secondly, the element *-? , that is, is replaced by the smaller element The condition that the curve is always to be convex down- wards has been adopted to render the problem simple and definite. If this condition be not attached we may have P changing sign in the course of the integration. Moreover P would vanish through- out any arc of the parabola which the particle might freely de- scribe under the action of gravity. 287. It may be useful to notice the formulae which we obtain when we generalise the preceding problem. Let u = fe< (p) ty (x) + px 0)] dx. Then for a maximum or minimum value of u we must have $ (p) ty'(x) % (a;) = a constant. The term of the second order in &u reduces to - \ JV the whole taken between the limits. 288. In the problem discussed in Chapter V. I arrived at the result that no valid objection could be brought from Jacobi's method against the conclusion that a minimum value of the sur- face had been obtained. I shall now make some more remarks respecting the application of Jacobi's method to such problems. 18 274 .MISCELLANEOUS OBSERVATIONS. When the value of y is fixed at the limits, we know by the investigation of Art. 23 that the term of the second order in the variation can be put in the form where X Q and x^ denote the limiting values of x. Separate the integral into two parts, one extending from x to f, and the other from to x^ Consider the former part; put Sy = tz, where z is such that In the second part of the integral put By = T where f also satisfies the same differential equation as z does. The two solu- tions z and f are of course not necessarily the same; they may differ by taking different values of the arbitrary constants which will occur. By thus separating the integral into two parts we shall find that the above term of the second order in the variation becomes The terms under the integral signs are positive if Q is posi- tive. The term outside the integral sign becomes by substitution that is, - this assumes that there is no discontinuity at the point for which x = g , so that Bp may have only one value at that point. MISCELLANEOUS OBSERVATIONS. 275 If then we can secure that Ql ^J is positive at this point, we are sure that the whole term of the second order in the varia- tion will be positive. Similarly we might proceed, if we wished to separate the in- tegral into more than two parts. Now let us apply this process to the problem of Chapter V. We shall separate the integral which represents the term of the second order in the variation into three parts. From the A. point A to some point between P and D we shall use the trans- formation By = tg ; and for z we shall take C^p, where O l is a constant. Then from the point thus reached to a second point between D and Q we shall use the transformation &/ = rf; and for fwe shall take C a pv t where C 9 is a constant, and v is the same as in Art. 103. And from the second point to B we shall use the transformation By=tz] and for z we shall take C 3 p, where C 3 is a constant. These transformations are legitimate ; for p does not vanish, except at D ; and the points between P and Q may be so taken that v does not vanish between them. Let f t denote the abscissa of the point taken between P and D, and let f 2 denote the abscissa of the point taken between D and Q ; then the term of the second order in the variation consists of parts under the integral sign, which are positive, since Q is positive, together with 276 MISCELLANEOUS OBSERVATIONS. Now 1-2, z p' f pv p v dx = q (!+/)*. p Zayvp* therefore i'' Thus, using the suffix 1 to apply to a value when # = , and the suffix 2 to apply to a value when x = i* z , we find that the term of the second order in the variation 4a"'" -f_ where the integral extends over the whole range ; from x^ to x = %^ we take rj =pv, and for the remainder of the integral we take i) = p. The part which is under the integral sign is positive ; but the other part is negative ; for we have in fact implied that v l is negative and that v z is positive. Thus we cannot assert that the whole expression is always positive. Nevertheless, it is obvious that we might in many cases conclude that the expression is positive. We know that y must vanish and change sign in the course of the integration ; see Art. 103. If By vanishes twice within the range which includes the values v l and v 2 , we may suppose these values to correspond to the cases ; so that Sj^ = 0, and y 2 = ; and then the expression is necessarily positive. [We may give another illustration of the process. Suppose that as p passes from its greatest value to zero v diminishes from the positive value X to zero, then becomes negative and passes MISCELLANEOUS OBSERVATIONS. 277 from zero to negative infinity ; when p changes sign v changes sign also ; see Art. 103. Let IJL denote a negative value of v, and suppose fi numeri- cally greater than X. Let us separate the integral into two parts, one extending from the initial value of x up to the value which makes v equal to //,, and the other from this value of w to the final value. In the first part of the integral we use the transformation Sy = iz y where z = G l p (1 + mv) ; and in the second part of the in- tegral we use the transformation By = r where f = O z p (1 + nv). It will be possible to give such values to the constants m and n, as to ensure that 1 + mv does not vanish within the first part of the integral, and that 1 + nv does not vanish within the second part of the integral. For these conditions will be satisfied if we take m positive and less than - , and n positive and between 1 , 1 - and - . X Then m n \dv mv 1 + nv dx 2ayp* ' and we find that the term of the second order in the variation becomes n-m (1 + mv} (1 + nv) yp* where the term free from the integral sign is to have the value corresponding to the value //, of v. In the term under the integral sign we take < rj=p(l+mv) for the first part of the integral, and 97=^(1 + nv) for the second part. The part which is outside the integral sign is negative. For n m is positive, 1 nip is positive, and I np is negative. 278 MISCELLANEOUS OBSERVATIONS. We see then that if fy vanishes for any point for which the value of v is numerically greater than X we may take //. to correspond to this point ; and then the term of the second order in the variation reduces to the positive part which is under the integral sign. The value of Q is - - 5 in the present problem; and by putting this value for Q some of the expressions will be simplified.] CAMBRIDGE : PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. PREPARING FOR PUBLICATION. A HISTORY OP THE MATHEMATICAL THEORIES OF ATTRACTION AND OF THE FIGURE OF THE EARTH FEOM THE TIME OF NEWTON TO THAT OF LAPLACE. EDUCATIONAL MATHEMATICAL WORKS BY I. TODHUNTER, M.A., F.R.S. Euclid for Colleges and Schools. New Edition. i8mo. cloth. 3*. 6d. Mensuration for Beginners. With numerous Examples. Second Edition. i8mo. cloth, is. 6d. Algebra for Beginners. With numerous Examples. New Edition. i8mo. cloth, is. 6d. Key to the Algebra for Beginners. Second Edition, Crown 8vo. cloth. 6s. 6d. 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