UC-NRLF 
 
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Geometric Properties Completely Characterizing 
 
 the Set of All the Curves of Constant 
 
 Pressure in a Field of Force 
 
 ' 4 I 
 BY 
 
 Eugenie M. Morenus 
 
 Submitted in partial fulfillment of the requirements for the degree of 
 
 Doctor of Philosophy in the Faculty of Pure 
 
 Science, Columbia University 
 
 PRESS OF 
 
 THE JOHN C. WINSTON COMPANY 
 
 PHILADELPHIA 
 
 I92Z 
 
. »   I . . . 
 
 Geometric Properties Completely Characterizing 
 
 the Set of All the Curves of Constant 
 
 Pressure in a Field of Force 
 
 BY 
 
 Eugenie M. Morenus 
 
 o^ 
 
 % 
 
 Submitted in partial fulfillment of the requirements for the degree of 
 
 Doctor of Philosophy in the Faculty of Pure 
 
 Science. Columbia University 
 
 PRESS OF 
 
 THE JOHN C. WINSTON COMPANY 
 
 PHILADELPHIA 
 
 1922 
 
CONTENTS 
 
 PAGE 
 
 Introduction v 
 
 Chapter I. Geometric Properties of the Set of oo * 
 
 Curves S c 1 
 
 Section 1. The differential equation of S c 1 
 
 2. Osculating parabolas 2 
 
 3. Hyperosculating circles 2 
 
 4. The focal circle 3 
 
 5. Correspondence between acceleration vectors and 
 
 element tangents 5 
 
 6. Characteristics of the quartic generated 6 
 
 7. The force vector 8 
 
 8. The radius of the hyperosculating circle 8 
 
 Chapter II. The Converse Problem for S c 10 
 
 Section 1. Nature of the Converse Problem 10 
 
 2. Converse of Property 1 10 
 
 3. Converse of Property 2 10 
 
 4. Converse of Property 3 11 
 
 5. Converse of Property 4 15 
 
 6. The parameter 16 
 
 Chapter III. Geometric Properties of the Quadruply 
 
 Infinite System 5 18 
 
 Section 1. Intrinsic equation of the system 18 
 
 2. The Cartesian equation 19 
 
 3. Osculating conies 20 
 
 4. Circles as curves of infinite pressure 20 
 
 5. Hyperosculating circles 21 
 
 6. The length of the force vector 22 
 
 7. The radius of curvature of conic 1 22 
 
 8. Second center of curvature of conic 1 23 
 
 9. Intercept of the asymptote of the cubic of Property 4 24 
 
 10. Direction of the cubic 25 
 
 1 1 . Hyperosculating parabolas 26 
 
 12. Intercept of the quintic of Property 7 28 
 
 (iii) 
 
 478713 
 
IV CONTENTS. 
 
 PAGE 
 
 Chapter IV. The Converse Problem for S 29 
 
 Section 1 . Converse of Property 1 29 
 
 2. Converse of Property 2 29 
 
 3. Converse of Property 3 29 
 
 4. Converse of Property 4 31 
 
 5. Converse of Property 5 32 
 
 6. Outline of final steps -. 33 
 
 7. Converse of Property 6 33 
 
 8. Converse of Property 7 34 
 
 9. Converse of Property 8 35 
 
CURVES OF CONSTANT PRESSURE 
 
 INTRODUCTION 
 
 In the Princeton Colloquium lectures, 1909, Professor Edward 
 Kasner of Columbia University pointed out several unfinished 
 problems connected with a field of force. He showed that the 
 trajectories whose characteristics he had previously described 
 (Transactions of the American Mathematical Society, Vol. 7, 
 No. 3, pp. 401-424, July, 1906) might be considered as a special 
 case of either of two more general problems : to find curves along 
 which a constrained motion is possible such that the pressure of 
 the moving particle against the curve is (1) proportional to the 
 normal component of the force or (2) constant. 
 
 The pressure, since the curve is considered smooth, is con- 
 nected with the normal component of acceleration by the formula 
 
 v 2 
 
 P = N. In the case of trajectories a particle moves freely 
 
 r 
 
 under the action of a force which depends only on the position of 
 the particle; that is, there is no pressure and P = o. P = o is ob- 
 tained when k = o from P = k N, which represents the first general 
 problem, or when c = o from P = c, which represents the second 
 general problem. 
 
 Regarding P = o as a special case of P = k N, Professor Kasner 
 stated five properties characterizing the system S* of <» 3 curves 
 corresponding to any value of the parameter k. Sarah Elizabeth 
 Cronin in her dissertation, 1917, found geometric properties 
 completely characterizing the system of °° 4 curves obtained by 
 combining all the systems Sjt. 
 
 It is my purpose to consider the problem represented by P = c, 
 the problem of curves along which a constrained motion is possible 
 such that the pressure against the curve remains constant. 
 
 I. I shall prove that the system S c of °o 3 curves of constant 
 pressure corresponding to any one value of the parameter c has 
 four properties. 
 
 Property 1. For any given lineal element (x, y, y') the foci 
 of the osculating parabolas of the single infinity of curves deter- 
 mined by the given element lie on a circle passing through the 
 given point. 
 
 (v) 
 
Vi INTRODUCTION. 
 
 Property .2. Of the <» l curves having the given lineal element 
 (x, y, y' ,) one has contact of the third order with its circle of 
 curvature. The locus of centers of the oo 1 hyperosculating circles 
 obtained by varying the initial direction is a conic passing through 
 the given point in the direction of the force acting at that point. 
 
 Property 3. The circle that corresponds according to Prop- 
 erty 1 to a given lineal element is so situated that the element 
 bisects the angle between the acceleration vector for the given 
 element and the tangent to the circle at the given point. 
 
 Property 4. That curve corresponding to a given lineal ele- 
 ment, which according to Property 2 has third order contact with 
 its circle of curvature has a radius of curvature equal to three 
 times the ratio of the tangential component of the force to the 
 normal component of the space derivative of the force. 
 
 II. In the second part it will be shown that the four proper- 
 ties described are sufficient to determine a set of oo 3 curves of 
 constant pressure in a field of force. 
 
 III. By varying the parameter c we get &> l sets of curves S c 
 and therefore oo 4 as the total number of curves of constant 
 pressure in a given plane field. The following properties of the 
 quadruply infinite system are found to be sufficient to completely 
 characterize it. 
 
 Property 1. Those curves of constant pressure which have 
 a given curvatiire element have osculating conies whose centers 
 lie on a conic tangent to the given element. 
 
 Property 2. The oo 3 circles of the plane are curves of con- 
 stant pressure and the pressure for these curves is infinite. 
 
 Property 3. The radius of curvature for the conic of Prop- 
 erty 1 is one-tenth the difference of the radius of curvature for 
 those curves having third order contact with their circles of 
 curvature minus the radius of curvature of the given curvature 
 element. 
 
 Property 4. As the curvature of a given element varies, 
 leaving the direction and point fixed, the second center of curvature 
 of the conic of Property 1 describes a cubic curve which passes 
 through the given point once and has an asymptote parallel to the 
 fixed direction with a double point on the asymptote at infinity. 
 
 Property 5. The asymptote of the cubic of Property 4 inter- 
 sects the normal of the lineal element to which the cubic belongs 
 
introduction! " , - •»' : ■-:■:: ;vii 
 
 at a point whose distance from the fixed point of the element is 
 three-tenths the ratio of the tangential component of the force to 
 the normal component of the space derivative of the force. 
 
 Property 6. The tangent of the angle which the cubic of 
 Property 4 makes with the normal to the fixed element is one-tenth 
 the sum of three ratios. The first ratio is negative three times the 
 product of the normal component of the second space derivative of 
 the force, as specialized for those curves which have third order 
 contact with their circles of curvature, multiplied by the tangential 
 component of the force divided by the square of the normal 
 component of the first space derivative of the force. The second 
 ratio is four times the tangent of the angle which the first space 
 derivative of the force makes with the normal. The third is the 
 negative tangent of the angle which the fixed element makes with 
 the force vector. 
 
 Property 7 . Of the oo l curves of constant pressure having 
 a given curvature element, two have contact of the fourth order 
 with their osculating parabolas. The locus of the foci of these 
 hyperosculating parabolas as the curvature varies, leaving the 
 lineal element fixed, is a quintic curve having a triple point at the 
 origin and having the element as double tangent. The third 
 branch of the quintic at the origin is so placed that its tangent at 
 that point makes with the line, of force an angle bisected by the 
 element. 
 
 Property 8. The quintic of Property 7 intersects the fixed 
 element tangent at a point whose distance from the point of the 
 element is five-halves the ratio of the normal component of the 
 space derivative of the force to the normal component of the 
 second space derivative of the force for those curves of constant 
 pressure which are hyperosculated by the particular parabola 
 whose focus is the point of intersection of the quintic and the ele- 
 ment tangent. 
 
 IV. Property 1 is characteristic of any set of oo < curves 
 represented by a certain general fourth order differential equation 
 and the other seven properties stated above are shown to be suf- 
 ficient to specialize the equation and identify a set of oo 4 curves 
 as curves of constant pressure. 
 
 The writer takes this opportunity to gratefully acknowledge 
 her indebtedness to Professor Kasner for his helpful criticisms 
 and suggestions. 
 

 CHAPTER I 
 
 Properties of the Set S c of °o 3 Curves of Constant 
 Pressure Corresponding to One Value of the Parameter c. 
 
 1. The differential equation of S c . 
 
 Given a plane field of force in which the general equations 
 of motion for a particle moving freely are 
 
 (i) m di^ =<p ( x *y)' m w = * <yX ' y ^' 
 
 Let a particle, whose mass may without loss of generality be 
 assumed to be unity, be projected at any time t = t from any posi- 
 tion x = x , y = y and with any velocity defined by the direction 
 y' = y' and the speed v = v . It is required to find a smooth curve 
 such that the pressure of the particle against the curve shall 
 remain constant. 
 
 Since the curve is smooth the tangential component of the 
 force vector 
 
 (2) T = v v g gives the acceleration a t of the particle along the curve. 
 
 The acceleration along the normal involves the constant pressure 
 P = c and is given by the formula 
 
 v 2 
 
 (3) a n = — = N+P where N is the normal component of the force 
 
 vector. It is evident that for a given value of the parameter 
 P = c the curve is uniquely determined. By varying the arbitrary 
 constants x , y , y'„, v we obtain in all, since each curve could be 
 described from any one of its points, oo 3 curves of constant pres- 
 sure in the plane corresponding to one value of c. Taking into 
 account the various values of c we find oo of these triply infinite 
 systems of curves S c . The system of trajectories forms one of the 
 triply infinite systems corresponding to c — o. 
 
 ... v 2 
 
 Eliminating v from the equations — = N+c and v v s =T gives 
 
 T 
 
 a differential equation in intrinsic form of the system S c . The 
 equation is 
 
 (4) (N+c)r s = 2T-rN s . 
 
 To find the equivalent equation in Cartesian coordinates the fol- 
 lowing substitutions are necessary : 
 
 (1) 
 
••■•.■■■ 
 
     - • . • . • . 
 ..."   • • 
 
 CURVES OF CONSTANT PRESSURE. 
 
 (5) N=+^L, T = ^M 
 
 _ [*«+(*,-*>,) y'-<p y y' 2 } (l+y")-y" (y+yV) 
 
 (7) r= (i+^!E )ra _3/y^-y"(i+/ 2 ) 
 
 r y" 2 
 
 The result is 
 
 (8) y „, = y" (ts-y'vs) ^i+y' 2 ) 3 y" 2 (c y - ^ V 1 +/*) 
 
 cVi+y2 + ^_y^ V1+/2 ( c Vi+ya+^-yV) 
 
 This is therefore the differential equation of S c . 
 
 2. Osculating parabolas. 
 Equation (8) has the form 
 
 (9) y'" = Gy"+Hy"K 
 
 It has been proved (Kasner, Trajectories of Dynamics, Transac- 
 tions of Amer. Math. Soc. 1906) that every system of curves 
 represented by an equation of this form has Property 1 of tra- 
 jectories. We may therefore state the following 
 
 Theorem I. If in any field of force a particle is projected from 
 a given point in a given direction, to each initial speed there 
 corresponds a definite curve along which the pressure of the 
 particle against the curve remains constant. The 00 l curves 
 of constant pressure obtained by varying the initial speed are so 
 situated that the locus of the foci of the osculating parabolas 
 constructed at the given point is a circle passing through the given 
 point. 
 
 This theorem expresses Property 1 for the set of 00 3 curves 
 of constant pressure S c . 
 
 3. Hyperosculating circles. 
 
 Of the 00 ' curves of constant pressure having the given lineal 
 element (x, y, y') it can be shown that one has contact of the third 
 order with its circle of curvature. 
 
 The curve and its circle of curvature must have the same 
 values of y', y" ', y'" at the chosen point in order to have contact 
 of the third order. The differential equation of all circles is 
 
 3 y'y' n 
 
 : i+y 
 
 (10) y"> 
 
CONVERSION OF PROPERTIES. 3 
 
 Placing this value of y'" equal to that in equation (8) for S c and 
 
 simplifying, we have y" = ~ ? , 7i? ', unless y" = o, which 
 
 3 {<p-\-y\f/) 
 
 leads to straight lines only and is therefore excluded from the 
 present discussion. This shows that for every value of x, y, y' 
 there is one and only one value of y" under the required con- 
 ditions. Therefore there is one curve of S r having third order 
 contact with its circle of curvature. 
 
 Moreover, this value of y" is independent of c and in fact is 
 exactly the value obtained in the corresponding discussion for 
 trajectories when the independent variable of the derivatives 
 has been changed. 
 
 The coordinates of the center of curvature for any curve are 
 
 -/ (1+y 2 ) v _i+/ 2 
 
 y" ' /' ' 
 
 Since y" is determined by x, y, y' for those circles having third 
 order contact and is the same for all systems S c , it is evident that 
 the center of the hyperosculating circle corresponding to a given 
 element is the same for all systems S c . 
 
 When y' varies the locus of the centers of hyperosculating 
 circles for S c will be the same as that found for trajectories, a 
 conic passing through the given point in the direction of the force 
 in the field (<p, yp). 
 
 Theorem II. In each direction through a given point there 
 passes one curve of constant pressure which has third order con- 
 tact with its circle of curvature. The locus of the centers of the °° l 
 hyperosculating circles obtained by varying the initial direction, 
 is a conic passing through the given point in the direction of the 
 force acting at that point. 
 
 Theorem II expresses Property 2 for 5 C . 
 
 4. The focal circle. 
 
 The equation of the focal circle corresponding by Theorem I 
 to a given lineal element for the general system (9) is 
 
 (11) 2G(a 2 +0 2 ) 
 
 +[3 jyi_l)_y (y*+i)H\a 
 +[(y'*+l)H-6y']p = o, 
 
 where a and /3 are coordinates of the focus of the variable osculating 
 parabola referred to the fixed point (x, y) as origin. Since in the 
 case of curves of constant pressure 
 
4 CURVES OF CONSTANT PRESSURE. 
 
 and 
 
 // = 
 
 Vi+y 2 (* Vi+ya+^-y^)' 
 
 the equation of the focal circle becomes 
 
 (13) 2 (*.-/<?.) Vi+yt («*+£=) 
 
 +3 [(t' 2 -1)JH-2 yV-c Vl+/2] a 
 
 + 3 [-c/ Vl+/ 2 +<p ty*- l)-2 /*] ^ = o. 
 
 Differentiating equation (13) with respect to a we find that the 
 direction of the circle at the point a = o, /3 = o is given by the slope 
 
 c Vl+/2-(/2-l) y/,-2 y'tp 
 
 ■cy' Vl+y a +(y 2 -l) <p-2y') 
 
 The tangent of the angle which this direction makes with the 
 chosen element is then found to be 
 
 If equation (13) is simplified by rotating the axes of coordinates 
 so that the X-axis coincides with the tangent at the fixed point, 
 thus making y' = o, we have instead of (13) 
 
 2 >£ s (a 2 +0 2 )-3 (Jt+c) ffl-3 <p(3 = o 
 
 and the tangent of the angle between the focal circle and the 
 
 element becomes the slope — — . 
 
 <P 
 
 In the case of trajectories, where c = o, this slope reduces to 
 
 \L . . . . 
 
 — , which is known to be the slope of the focal circle corresponding 
 
 to the system of trajectories which pass through the given point 
 in the given direction. 
 
 Returning to the components of acceleration of the particle 
 moving along the curve of constant pressure, a x =T, a n = N+c, 
 the corresponding components along the axes are 
 
 a x =T cos 6— (N+c) sin 
 a u =T sin + (AT+c) cos 
 
CONVERSION OF PROPERTIES. 
 
 where d = tan' l y'. Hence the slope of the acceleration vector 
 
 ay _ xP Vl+/ 3 +c 
 a x <p Vi-j-y2_ 6 y ' 
 
 Transforming this expression by rotation so that y' = o it becomes 
 - — . Since the slope of the tangent to the focal circle has been 
 shown to be   — we have proved 
 
 Theorem III. The circle that corresponds according to 
 Theorem I to a given lineal element is so situated that the element 
 bisects the angle between the acceleration vector for the given 
 element and the tangent to the circle at the given point. 
 
 This constitutes Property 3 for S c . 
 
 5. The correspondence between the direction of the accelera- 
 tion vector and that of its corresponding element. 
 
 We have seen that for every given element there is a cor- 
 responding focal circle whose direction at the given point is repre- 
 sented by the slope 
 
 _ c ^i+y / 2 -(/ 2 -i) * - 2-yV 
 -cy' Vl+y 2 +(/ 2 -l) <p-2 y'+ 
 
 Also for every element there is a corresponding acceleration vector 
 for the particle moving along the curve of constant pressure which 
 has the given element. The slope of the acceleration vector is 
 
 j, Vi+ys-fc 
 
 — , . There are thus two pencils of lines through the 
 
 given point besides the pencil formed by the tangent of the given 
 element as it revolves about the fixed point. Moreover, the 
 pencils are so related that the tangent of the given element always 
 bisects the angle formed by the lines corresponding to it in the 
 other two pencils. A study of these pencils furnishes an inter- 
 esting description of them. 
 
 When the pencil of tangents represented by the slope y' is 
 moved to a separate point and the locus of its points of intersection 
 with corresponding rays in the pencil of acceleration vectors is 
 traced, the locus proves to be a quartic curve of a distinct character. 
 Because this curve provides a means of describing the pencil' of 
 acceleration vectors in a purely geometric way which is useful in 
 the converse problem to be considered later, its generation will 
 be given somewhat in detail. 
 
6 CURVES OF CONSTANT PRESSURE. 
 
 A simple form of the equation of the locus is found by taking 
 the first chosen point (x, y) as origin and placing the center of 
 the pencil of element tangents at a point whose coordinates are 
 a <p and a \p with reference to the chosen point as origin, i. e., at 
 any point on the line of force. The pencil of lines at the origin 
 
 y \l/ Vi -|_3, / 2_j_ c 
 
 is then represented by - = ,   , while the pencil at 
 
 . y — a. ib . . . . 
 
 {a <p, a \f/) is =y , corresponding lines being those which m- 
 
 %—a <p 
 
 volve the same value of y' . 
 
 Eliminating y' from these equations we have the equation of 
 
 the locus of intersections of corresponding lines. It is found to be 
 
 (14) (v? y-yf, x) 2 [(x-a <p) 2 + (y-a r/0 2 ] 
 = c 2 [x (x — a <p)-\-y (y — a i^)] 2 . 
 
 The conclusions reached above may be stated as 
 
 Theorem IV. The correspondence between the elements at 
 a given point and the acceleration vectors at that point is such 
 that when the first pencil of rays is moved to a point on the line 
 of force the locus of points of intersection of corresponding rays is 
 a quartic curve. 
 
 6. Distinguishing characteristics of the quartic in Theorem IV. 
 
 The equation of the quartic of Theorem IV can be somewhat 
 simplified by choosing the center of the pencil of element tangents 
 at the point (<p, \p) thus making the distance between the centers of 
 the pencils represent the magnitude of the force vector "v<p 2 -j-^ 2 . 
 The equation of the quartic is then 
 
 (15) {.<py-txY[{x-vY+(y-W\ 
 
 = c 2 [x (%—ip)+y {y-i)Y- 
 
 When this equation is differentiated and the slope of the curve 
 
 / dv\ 
 
 found at the point (<p, \p) we get I <p-\-yp -f- ) 2 = o. 
 
 Therefore the curve has a double tangent at (tp, \p) and the tangent 
 is perpendicular to the line of force. 
 
 In a similar way, there are found to be two distinct tangents 
 at- the origin and their slopes are 
 
 dx ' ^> 4 + < pV 2 -cV 2 
 
CONVERSION OF PROPERTIES. 7 
 
 The tangents of the angles which these lines make with the line 
 
 of force - = - are then found to be , i. e., they are pro- 
 
 x <p ± yJ^-\-^ 
 
 portional to the positive and negative reciprocals of the length of 
 the force vector, and the constant of proportionality is the param- 
 eter of the system of curves of constant pressure. It is evident 
 also that the 'line of force bisects the angle between the two 
 branches of the quartic. The facts given above form 
 
 Theorem V. The quartic curve of Theorem IV has a double 
 tangent perpendicular to the line of force at the center of the 
 pencil of element tangents and two distinct branches which make 
 equal angles with the line of force at the center of the pencil of 
 acceleration vectors. The tangents of the equal angles just men- 
 tioned are proportional to the positive and negative reciprocals of 
 the length of the force vector, the constant of proportionality 
 being the parameter of the system of curves of constant pressure. 
 
 Rotate the axes and translate them so as to place the tac-node 
 of the quartic at the origin with the x-axis as the double tangent 
 
 and the crunode at the point (o, ^<p 2 -\-$ 2 ). The y-axis then co- 
 incides with the line of force and the equation has the form 
 
 x 2 (<p 2 +yfr 2 ) (x 2 -\-y 2 )=c 2 {x 2 +y 2 -y Vp*+f*)». 
 
 It is apparent that the curve is now symmetrical with respect to 
 the j-axis which is the line of force. By differentiating it is found 
 that at the origin 
 
 dx 2 " 
 
 L v+* 2 d " c \ 
 
 The curvature of the two branches is thus seen to be the same as 
 the curvature at the vertex of the two parabolas 
 
 y = 
 
 , l + -1 x 2 and y = [ . l - -1 x 2 
 
 respectively. Since the constant coefficient in these equations 
 gives the reciprocal of the latus rectum of the parabola the latus 
 
 rectum of the first is — - — and of the second 
 
 c+ V^+^2 c _ V^2 + ^2- 
 
 When the pencil of element tangents is at any other point on 
 the line of force (a<p, a\p) the directions of the tangents mentioned 
 in Theorem V remain unchanged but the curvature of the two 
 
8 CURVES OF CONSTANT PRESSURE. 
 
 branches at the tac-node vary with a, for -j—„ = - — -. =*= - • 
 
 dx 2 a [ V^2_)_^2 c J 
 
 The length of the force vector V^> 2 -}-^ 2 is thus identified as that 
 distance between the centers of the generating pencils of rays 
 which produces the simplest expression for the curvature of the 
 branches of the quartic at the tac-node. 
 
 Theorem VI. The quartic of Theorem IV is symmetrical 
 with respect to the line of force. When the pencil of element 
 tangents is on the line of force and at a distance from the fixed 
 point of the chosen element equal to the length of the force vector 
 the two branches of the quartic at the tac-node have the same 
 curvature as that at the vertices of two parabolas : the latus rectum 
 of one parabola is equal to the product of the parameter of the 
 system of curves of constant pressure multiplied by the length of 
 the force vector divided by the sum of the parameter and the 
 length of the vector, while the latus rectum of the other parabola 
 is the product of the same parameter and length divided by their 
 difference. 
 
 7. Geometric construction of the force vector. 
 Properties 2 and 3 furnish sufficient information for the 
 
 geometric construction of the force vector. The direction of it is 
 given by the direction of the conic which according to Theorem II 
 is the locus of the centers of the hyperosculating circles belonging 
 to the curves of constant pressure passing throtigh a given point. 
 The magnitude of the force vector is represented by that distance 
 between the centers of the pencils of rays generating the quartic 
 curve of Property 3 which produces the simplest expression for 
 the curvature of the branches of the quartic at the tac-node. 
 
 8. The curve which has a hyperosculating circle. 
 Returning to Property 2 we see that the curvature of the 
 
 curve of constant pressure corresponding to a given lineal element 
 which has third order contact with its circle of curvature is char- 
 acterized by the second derivative 
 
 . „ = (*.-yV.) (l+y'T , 
 y 3 {ip+y'f) 
 
 Hence its radius of curvature 
 
 3 (y+yW 3 T 
 
 f*-y<t>8 n 
 
 where ]v represents the normal component of the space derivative 
 
CONVERSION OF PROPERTIES. 9 
 
 3 T 
 
 of the force vector. The fact that R c = -=- constitutes the fourth 
 
 N 
 
 property for S c and may be stated as 
 
 Theorem VII. That curve, corresponding to a given element, 
 which according to Theorem II has third order contact with its 
 circle of curvature has a radius of curvature equal to three times 
 the ratio of the tangential component of the force to the normal 
 component of the space derivative of the force. 
 
CHAPTER II 
 The Converse Problem for 5,.. 
 
 1. Nature of the converse problem. 
 
 It will now be shown that the four properties found for 5, 
 are sufficient to identify a triply infinite system of curves as curves 
 of constant pressure in a field of force. It will appear that curves 
 having Property 1 are represented by a general differential equa- 
 tion of the third order having two undetermined coefficients, 
 functions of x, y, y' and that each of the other properties specializes 
 the coefficients until the equation (8) is developed. 
 
 2. Converse of Property 1. 
 
 Professor Kasner has proved that the only triply infinite 
 systems of curves possessing the property that for every lineal 
 element the corresponding focal curve is a circle passing through 
 the point of the element are those defined by a differential equation 
 of the form 
 
 (9) y'" = G (x, y, f) y"+H (x, y, y') y"* 
 
 where G and H are arbitrary functions of x, y, y'. This type 
 includes trajectories and all other curves of constant pressure 
 as special cases since G and H for these curves have a particular 
 form. 
 
 3. Converse of Property 2. 
 
 A restatement of Theorem II avoiding the words of dynamics 
 is as follows : Of the o° l curves associated with a given lineal ele- 
 ment one has contact of the third order with its circle of curvature. 
 The centers of the o° x circles, obtained by varying the initial direc- 
 tion, lie on a conic which passes through the given point in a fi^ed 
 direction whose slope is co. 
 
 The general equation of a conic passing through the given 
 point in the direction w is 
 
 (16) X A' 2 -m X Y+v F 2 +3 (F-« A") =o. 
 
 Here A' and \ ' are referred to the given point as origin and X, \i, v are 
 any functions of x and y. 
 
 If this conic is to be the locus of centers of curvature it must be 
 
 — y> (\A-y'2) 1+/ 2 
 
 satisfied bv A = — .. — , Y = £- in which y and y belong 
 
 •> y" y> 
 
 to the curves whose osculating circles have centers A' and Y. 
 Substituting in (16) and simplifying, 
 
 ( 10) 
 
CONVERSION OF PROPERTIES. 1 1 
 
 -(1+/ 2 ) (X /»+/*/+*) 
 
 n 
 
 (1?) y " 3U+.J0 
 
 a relation which must be satisfied by X, n, v when 7' and y" belong 
 to a set of curves having Property 2. The condition that curves 
 of set (9) have third order contact with their circles of curvature 
 gives 
 
 (18) G+H y>' = \^ 
 
 since /" = ] 1? y * for circles and /" = G y"+H y" 2 for sets 
 
 \-\-y * 
 
 of curves having Property 1. 
 
 Substituting (17) in (18), that is combining Properties 1 and 2,- 
 
 (19) G = 
 
 3/ 1 f_ (!+/') (\/ 2 +m/ + »0 1 
 
 + i+y 2 JL 3(i+co/) J 
 
 Replacing G by this value in equation (9) we have the equation 
 of curves with Properties 1 and 2. 
 
 r-[- B ^][-^ 
 
 ){\y'--+ny'+v) 
 
 3(1+0;/) 
 
 /'+ H y' 
 
 4. Converse of Property 3. 
 
 To avoid the assumption of the existence of a field of force 
 Property 3 may be restated employing the facts of Theorems 
 IV, V and VI. These theorems enable us to describe the line 
 which forms with the tangent to the focal circle corresponding to 
 a given element the angle which is bisected by the given element 
 without using the fact that that line is the acceleration vector. 
 
 Property 3 is then: There exists for each element tangent 
 y' at the point (x, y) a certain line through the point such that the 
 angle between this line and the tangent to the focal circle cor- 
 responding to the element (x, y, y') is bisected by the element. 
 The direction of this line varies with the direction of y' forming 
 a pencil of lines C\ through the point (x, y) such that if the pencil 
 formed by /, C2, is moved to a point on the fixed line of slope o> 
 the points of intersection of corresponding rays lie on a quartic 
 curve. The quartic has a tac-node at C 2 with the double tangent 
 perpendicular to y' = o> and two distinct branches at d forming an 
 angle bisected by y' = w. Moreover, there exists a vector F in 
 the direction y' — w such that the tangents of the angles which 
 the branches of the quartic at C\ make with y' = co are proportional 
 to the positive and negative reciprocals of the magnitude of the 
 
12 CURVES OF CONSTANT PRESSURE. 
 
 vector, the constant of proportionality being an independent 
 parameter. The quartic curve is symmetrical with respect to the 
 line y' = <j). When C 2 is placed so that the distance C\ C 2 = [F], 
 the curvatures of the two branches at C 2 are the same as those at 
 the vertices of two parabolas: the latus rectum of one parabola 
 is the product of the constant proportionality factor mentioned 
 above and the distance between the centers of the pencils divided 
 by the sum of the same constant and distance, while the latus 
 rectum of the other parabola is the product of the constant pro- 
 portionality factor and the distance between the centers of the 
 pencils divided by the difference of the same constant and distance. 
 
 The information thus expressed is sufficient to determine the 
 direction of the line corresponding to any value of y' and therefore 
 of the tangent to the focal circle corresponding to a given element. 
 A demonstration of the proof of this statement follows. 
 
 Let the fixed length of the vector F described above be marked 
 off from the chosen point (x, y) in the fixed direction y' = co and 
 let its projections on any pair of rectangular axes through (x, y) 
 be <p and \f/. <p and \p are thus arbitrary functions of x and y in 
 that the axes are arbitrary but, once the axes are placed, ip and \p 
 are determined by the length of F which may now be expressed 
 
 V+V' 2 - 
 
 In the dissertation of Ruth Gentry, Plane Quartic Curves, 
 1896, the general equation of a quartic having a tac-node is given 
 in trilinear coordinates: 
 
 (yz — mx 2 ) (y z — m'x 2 ) —2 c x y 2 z-\-f x ?j y-\-gx 2 y 2 -\-dy 3 z+hx y 3J rl y x 
 and the tac-node is at (x, y). 
 
 When this is changed to rectangular coordinates and the terms 
 containing odd powers of x dropped to make the curve symmetrical 
 with respect to the j-axis it becomes 
 
 (20) (y — m x 2 ) (y — m'x 2 ) =g x 2 y 2 +d y 3 -\-l y 4 . 
 
 The coefficients m, m' , g, d and / will now be determined by means 
 of the facts under Property 3. 
 
 Bv differentiation it is evident that : - and — -. are the latus 
 
 m m 
 
 rectums of parabolas having at the vertex the same curvature as 
 
 the two branches of the quartic (20) at the origin. Hence for 
 
 the quartic of Property 3, 
 
 c+ V+f- c - V^+^ 
 
 m = — , and m = — , . 
 
 c ^<p 2 +4< 2 c V^+^ 
 
CONVERSION OF PROPERTIES. 13 
 
 Substituting x = o in (23), y 2 —dy 3 — ly i = o. y 2 = o indicates 
 the double point at the origin and ly- — dy—l=o, two other 
 intercepts on the y-axis. In order that these two may coincide, 
 placing the second double point on the j-axis, the roots of this 
 
 JO 
 
 equation must be equal, i. e. , d- — 4 I = o. Hence I = — — . Solving 
 
 2 
 the quadratic then y = — -,. This value corresponds to the distance 
 
 between the centers of the pencils in the quartic of Property 3, 
 
 VJi+ji Hence d = ~^^ 2 and / = - ^L_ 
 
 Differentiating the general quartic (20) we find that, at the 
 
 . / 2\ dy 1 , 
 
 point (°'w)'/f :=± 5^ 2wd + 2 *»'d+4 g. Therefore the tangents 
 
 to the quartic at the point \o,-%\ make with the v-axis angles whose 
 
 tangents are ± ,- j— — — Tmr- • These values correspond to 
 
 V2 m d+2 md+4 g 
 
 the tangents of the angles which the two branches of the quartic 
 
 in Property 3 make with the line joining the centers of the pencils. 
 
 Therefore ± r~r^ — , , , A — = + 
 
 ^ 2 m d-\-2 w'd-f-4 g — >l<p 2 -\-\f/' y 
 
 Substituting m— , 
 
 c V+* 2 
 
 and solving for g. 
 
 g = 
 
 m = , 
 
 C V^+^2 
 
 V+* 2 
 
 ^2 + ^2_2 C 2 
 
 * 2 +v 
 
 Now, replacing hi, m', d, g and I by their values the general quartic 
 
 becomes, simplified, 
 
 * 2 (*-++*') 0t-- + v 2 )=c 2 (x*+y*-y V^+^«)«. 
 
 By translation and rotation the axes may be placed so that the 
 
 point (p, o) becomes (<p, \[/) and (o, ^<p 2 +\p-) is the origin. The 
 
 equation of the quartic is then (15). 
 
 v — ^ 
 
 Now a pencil of lines through (<p, \p),y' = : -, should intersect 
 
 x — <p 
 
 this curve in points such that the lines joining the points to the 
 
14 CURVES OF CONSTANT PRESSURE. 
 
 origin will be the lines described in Property 3, i. e., the accelera- 
 tion vectors corresponding to the directions of the elements 
 through the chosen point (%, y). Substituting y = \[/-\-y' {x — <s) in 
 the quartic equation (15) gives the factor (x — <p) 2 = o representing 
 the abscissa of the double point at (<p, \J/) and 
 
 (+-y'<p) (ip Vl+y /2 -c/) 
 VT+7" 2 ty-y'<p+c Vi+ya 
 
 or 
 
 Hence 
 
 y =4>-\-y' (x— <p) = 
 
 Vi+/ 2 {yp-y'ip-c Vi+/2) 
 
 (*-?V) (* VT+72+c) 
 
 Vl+y'2 (^- 3 ;V+C V 1+y, 2 ) 
 
 ty-yV) (^ Vl+y 2 -c) 
 
 Vi+y 2 (^-tV-c v 1+3/2) 
 
 The equation of the pencil of lines passing through the origin and 
 the points indicated by the pairs of coordinates just found is 
 
 y _ ±i£ Vl+y 2 + r 
 
 x ±^ Vi+y 2 - c y' 
 
 The ambiguous sign is accounted for by the fact that Vl-f-y' 2 
 represents the secant of the angle whose tangent is y' and its sign 
 depends on the quadrant in which tani'V lies. Corresponding 
 to any value of y' then we have a line through the origin satisfying 
 the description in Property 3 of the acceleration vector. 
 
 The converse problem is now to find those systems of curves 
 (9) in which each focal circle makes with its corresponding element 
 an angle equal to the angle which the element makes with a certain 
 line whose direction is determined by the element and whose slope is 
 
 = * Vl+/ 2 +c 
 " p Vi+/ 2 -c/ 
 
 The slope at the given point of the circle (11) belonging to the 
 
 general system (9) is ^y _ {(l + y >^ H   
 
 Hence the tangent of the angle which the focal circle makes with 
 
 3 
 
 the element is — 
 
 3 /-(!+/») H' 
 
CONVERSION OF PROPERTIES. 15 
 
 By the condition imposed in Property 3 this must equal the 
 tangent of the angle which the element makes with the line of 
 slope a. 
 
   -3 y'-a- 
 
 ' '3/-(l+/') H ~~~' 1+ a y r 
 
 3 
 
 Solving for H we find H = —, . Hence the onlv svstems of oo 3 
 
 y —a 
 
 curves having Properties 1 and 3 are those whose differential equa- 
 
 3 
 
 tions are of the form V" = G y" -\ — : v" 2 , where G is anv func- 
 
 y —<y 
 
 j, Vi+y2_|_ c 
 tion of x, y, y' , and a = . The form of (9) when a is 
 
 <p vl+yt—cy' 
 substituted is 
 
 (21) y"'=Gy"+ 3( c y-^i+y 2 ),- 
 
 v'l+y 2 (c ^l+y' 2 +\l/-y'<p)' 
 
 Now replacing G by its value from (19) including the value of H 
 we have the differential equation of curves having the Properties 
 1, 2, 3 
 
 (92) y"> = (M' /2 +m/ + *)^ yt , 
 
 c <l-\-y' 2 +xP-y'<fi ' 
 
 3(n/-,pVl+/ 2 ) „ 2 
 
 V1+/2 (c vi+ya+^-y^) y 
 
 5. Converse of Property 4. 
 
 Restating Property 4 using the vectors mentioned in a purely 
 geometrical way we have : The radius of curvature for that curve 
 having the lineal element (x, y, y') which according to Theorem II 
 is hyperosculated by its circle of curvature is three times the ratio 
 of the projection of the vector F described in Property 3 upon the 
 tangent to the curve divided by the projection of the space de- 
 rivative of that vector upon the normal. This gives 
 
 R = 3 T _ 3 {<p+y'+) 
 
 (23) .-. y 
 
 n 
 
 (i+/ 2 r(*.-yV.) 
 
 3 (<?+/*) 
 
 Now the curves represented by (22) have third order contact with 
 their circles of curvature provided (23) is true, i. e., 
 
16 CURVES OF CONSTANT PRESSURE. 
 
 y>» = 3yy 
 
 (1+/ 2 ) 
 
 {\y'*+ny'+v) <py" 3 {cy'-<p Vi+y«) /' 2 
 
 if we substitute for /' the value given by (23). Making the sub- 
 
 to 
 
 stitution, replacing w bv - and simplifying the result we find 
 
 . ,, , , ,   Vi+/»(*.-yV.) 
 
 A V — |-A* V + " = " ~~ "• 
 
 Hence the requirement that the curves of equation (9) have 
 
 Property 4 is equivalent to the substitution of : — : — — 
 
 for the quadratic expression X y' 2 +M y' -\-v in the equation (9) as 
 already modified by Properties 1, 2, 3, that is in equation (22). 
 The equation is then restored to the special form (8). 
 
 6. The parameter in the equation. 
 
 It remains to show that this equation can represent no other 
 curves than curves of constant pressure in the field of force dp, \p). 
 The existence of the field of force has been established when the 
 functions <p (x, y) and \f/ (x, y) were found for every point (%, y) 
 in the plane. It must now be proved that the parameter c can be 
 accounted for only by a constant normal component of acceleration 
 compelling the particle in the field to move along the curves whose 
 characteristics are being studied. 
 
 Equation (8) may be written 
 
 yl „ = y" [(*.-/».) (l+f)+3f ( c y-gV !+/»)] 
 Vl+/ 2 ( € Vi+y 2 +^-yV) 
 
 Observing the form of the denominator of the right hand member 
 of the equation, it appears that when, at a given point, y' is such 
 
 that c ^l-\-y' 2 +\f/ — y'<p = o. it can be proved that the curves of (8) 
 are straight lines. This condition is equivalent to c+N = o. 
 Excluding the discontinuous case y'" = <» , 
 
 the numerator 
 
 y"l(*.-yV.) U+y' 2 )+3y" (c/-* Vi+/*)]=o. 
 
 Hence y" =0 
 
 3 (<:/-«> Vl+/») 3T V • 
 
CONVERSION OF PROPERTIES. 17 
 
 The first possibility makes y" f = o directly and therefore succeeding 
 derivations are zero and the curves are straight lines. 
 
 N f 
 
 If y = — — y (l+y' 2 ) consider the space derivative of the 
 
 normal component of force. 
 
 Since N = — c, N, = o. 
 
 — T . . 3 7 
 
 But N, = N and r for this value of y" is equal to -=-. 
 
 r N 
 
 .   . N, = N - y = f N. Since N t = o, N = o. 
 
 Consequently y" = o, y'" = o . . . 
 
 Thus both possibilities lead to the conclusion that at every point 
 when the slope is such that N+c = o the curves are straight lines. 
 Since the particle can move in a straight line only when it is affected 
 by a tangential component of acceleration alone, it is evident that 
 c is a normal component of acceleration which, when the element 
 tangent is in this direction, cancels the normal component of the 
 force (<p, \f/) and renders the normal acceleration of the particle 
 zero — that is, can be nothing but a constant pressure between 
 the particle and the smooth curve. 
 
CHAPTER III 
 
 Properties of the Quadruply Infinite System of All 
 Curves of Constant Pressure in a Plane Field of Force. 
 
 1 . The intrinsic equation for the system of » 4 curves. 
 
 Since one set of oo 3 curves S c , corresponding to one value of 
 the parameter c, is represented by the intrinsic equation (4) 
 (N-\-c) r s = 2 T — r N s , if we differentiate this equation and elim- 
 inate c we shall have a differential equation of the fourth order 
 representing the °o 4 curves of constant pressure in the field of 
 force (<p, yp). 
 
 The space derivative of the force, regarded as a vector, will 
 be used here. The normal and tangential components of this 
 vector are 
 
 (24) n = ^ s ~ y,<ps = ^*+(^y-p*) y'-<Pv y' 2 
 
 VI+/ 2 " l +/ 2 
 
 j = <P,+y'iPs = <px+(<py+tx) y'+Tpyy" 1 
 
 Vi+y 2 " 1+y 2 
 
 These components are related to the space derivatives of the 
 components of force in this way: 
 
 — T - N 
 
 (25) Ns=N--,T. = T+-. 
 
 r r 
 
 The space derivative of N also will be necessary. 
 
 (26) N e = N l -\--- where 
 
 r 
 
 Ni = yp xx -\-(2 yp xv -Vxx) / + 0/^-2 ip xu ) y" 2 -<Py y y' 3 
 = yp y-2 (<Py+ypx) y'+ivx-^y) y' 2 -<p* 
 
 . 2 ~ 1+y 2 
 
 Using (25) the equation (4) becomes 
 
 (N+c) r s = 3 T-Nr. 
 Differentiating with respect to 5 and eliminating c we obtain the 
 intrinsic equation. 
 
 (27) (Nr-3 T) rr s = (2 rN-T) r s 2 
 
 +[Ni r 2 +(/V 2 -3 T) r-3N]r, 
 which represents the <» 4 curves of constant pressure. 
 
 ( 18) 
 
CONVERSION OF PROPERTIES. 
 
 19 
 
 2. The equation of the system in Cartesian coordinates. 
 
 In order to investigate the characteristics of the system, 
 equation (27) may be translated into the corresponding Car- 
 tesian equation. The values of N, r, r s , T, Ni, N 2 and N have 
 already been mentioned in (24), (7), (5), (26). 
 
 (28) r„ = 
 
 y » % (3y „2 _ 2y , y „, ) _ y, y i v ( x +y >2 )+2 y'">{ 1 +/ 2 ) 
 
 y" 3 Vi+/ 2 
 
 After making these substitutions and simplifying, the equation 
 (27) becomes 
 
 (29) y IV [3 y" (<p+y'+)- {&+&,- *J y'-^y' 2 ] (i+/ 2 )] 
 
 = y'" 2 5 (*+/*) 
 
 +/ 
 
 +9y 
 
 ,"4 
 
 y" 2 (+-y'<p) 
 \+y' 2 
 
 -y"[lOy'i+ x +(+v-<Px)y'-<Pvy' 2 } 
 
 ++ y (i-4/ 2 )+^ (y 2 -4)-5 / (<p„+* z )] 
 
 -(i+y 2 )i^+(2 + xy -<p xx ) y+^yy-2 <p xy ) y'"- 
 -<py y y' 3 \ 
 
 <p 
 
 \+V 2 
 
 + 
 
 3y //3 r(5/ 2 -i)i^+(^-^)y-^/ 2 i 
 
 +3y'y" r ti'xx+(2t xy -<p xx ) y'+(fyy-2 <p Iy ) y' 2 -<Py y y' s } 
 
 To simplify the writing of this equation let 
 
 (30) ?!-*.+ (*,-*) /-W^ 1 
 
 P 2 = ^(l-4/ 2 )4-^ x (/ 2 -4) -5/(^4-^) 
 
 P 3 = ^xx+(2 *„-*«) /+(*»»- 2 *„) / 2 -^ v9 / 3 
 
 D = 3y / (^+>'V)-(14-/ 2 )P 1 . 
 We then have 
 
 (31) T iv = 5(^+yV) y , 2 
 
 D 
 
 /" r-3(^-y 
 
 z?(i+/ 8 )|_+ (i+/ 2 
 
 -3 (+-y'<p) y /j 4-(i+y«) (io yPi+p 2 ) y" 
 
 ) 2 Pa 
 
 "4 
 
 + 
 
 9yy 
 D(i+/ 2 ) 
 
20 CURVES OF CONSTANT PRESSURE. 
 
 s [(s j^-d ft+/fti /*• 
 
 + 
 
 D(l+/ 2 ) 
 
 3 y'P 3 ..,„ 
 
 It should be noticed that Pi, P 2 , P 3 are functions of x, y, y' but D 
 involves also y" . 
 
 3. Osculating conies. 
 Equation (31) is of the form 
 
 (32) jr*rAy"'*+B?"+C. 
 
 Professor Kasner has proved (Bulletin Amer. Math. Soc, March, 
 1907, vol. 3, p. 290) that the system represented by such an equa- 
 tion has the characteristic property that the curves having a given 
 curvature element (x, y, y', y") have osculating conies whose 
 centers lie on a conic tangent to the element. 
 We may therefore state 
 
 Theorem VIII. Curves of constant pressure having a given 
 curvature element {x, y, y\ y") have osculating conies whose 
 centers lie on a conic tangent to the given element. 
 
 This will be known as Property 1 for the quadruply infinite 
 set of curves represented by equation (29). The quadruply 
 infinite set of curves will hereafter be called 5. 
 
 4. Circles as curves of infinite pressure. 
 
 Considering the possibility that some of the °° curves having 
 a given curvature element might have contact of the third or 
 fourth order with their circles of curvature, those curves would 
 have to agree with circles in derivations up to the third or fourth 
 derivative. 
 
 3 y'v" 2 3 y" z (5 v' 2 +D 
 
 (33) For circles y'" = , , ,_ and v IV = — ,. . ' ,' n • 
 
 1+7" (1 +T ) 
 
 3 VV' 2 
 Substituting /" = \_ ,, in (31) we find 
 
 (%L . 1V 3 v"> (5 y'g+1) [3 y" (y+yV)-Pi (l+/ g )] 
 
 ( } y (1+y 2 ) 2 [3 y" {*>+?*) -Pi d + v' 2 )] 
 
 If 3 y" (tp-\-y'^) — P\ (l+y" 2 )^o that factor may be cancelled, 
 
 3 v" 3 (5 V 2 + 1 ) 
 y lv = , v ./, — - and for every value of v" all the curves have 
 
 (l+y 2 ) L 
 fourth order contact with their circles of curvature. In fact, the 
 
CONVERSION OF PROPERTIES. 21 
 
 curves are then circles since succeeding derivatives for them must 
 be obtained from y'; y", y'", y lv . 
 
 Referring to the third order problem of S c and solving equa- 
 tion (8) for c, we find 
 
 [P, y"-3 y yi-V" (^-vV)] Vl+/ 2 
 C v'"(1+t'-)-3 f y" 2 
 
 3 y'y'" 1 
 If y'" = . . >o > as it must if the curves are circles, the denomina- 
 \+y- 
 
 tor of the above fraction becomes zero and the numerator, 
 y" [3 y" (^+/^)-Pi(14\V 2 )]. It is evident then that c= oo 
 when the constant pressure curves are circles unless y" = o (when 
 the curves are straight lines)) or 3y"(<p+y'ip) — Pi(l+r' 2 ) =o 
 which is known to be the condition for third order contact with 
 circles in every S c . S x then consist of co 3 circles and these are 
 included among the °° 4 curves that satisfy equation (31). 
 
 Theorem IX. The oo 3 circles of the plane are included among 
 the curves represented by the given fourth order differential 
 equation. The pressure of the moving particle against these 
 circles is infinite. This theorem expresses Property 2 for the 
 system 5. 
 
 5. The hyperosculating circles. 
 
 If 3 y" {<p+y'\P)-P x (1 +/ 2 ) = o, or briefly D = o, we have 
 
 (35) ;; = 3^+7*T 
 
 It is known that in every S c the curve corresponding to a given 
 lineal element and having the curvature represented by (35) has 
 third order contact with its circle of curvature. Hence of the oo 2 
 curves in 5 corresponding to a given lineal element the circles of 
 curvature of oo > curves, one from each S c , coincide. As the direc- 
 tion of the element varies the center of this hyperosculating circle 
 describes a conic passing through the given point in the direction 
 of the line of force. This is the conic of Property 2 for every S c . 
 It is also known from the third order problem that the radius of 
 the hyperosculating circle mentioned above corresponding to a 
 given lineal element is three times the ratio of the tangential 
 component of the force to the normal component of the space 
 
 3 T 
 derivative of the force. R, = -^=^. 
 
 N 
 
22 CURVES OF CONSTANT PRESSURE. 
 
 6. The length of the force vector. 
 
 In order to find geometrically the length of the force vector 
 in the fourth order problem it is necessary to investigate the 
 relation of the circles of Property 2 in 5 to the quartic of Prop- 
 erty 2 in S c . The acceleration vectors when the constant pressure 
 is infinite fall along the normals. Consequently the points of 
 intersection of these lines with the corresponding tangents when 
 the pencil of tangents is moved to the point Ci at the extremity 
 of the force vector lie on a circle. This circle is the limiting form 
 of the quartic in S c as c approaches infinity and its diameter is 
 the required length of the force vector. 
 
 Theorem X. The °° 1 acceleration vectors for the circles of 
 Theorem X which pass through a given point are perpendicular 
 to the tangents to the circles at that point. When the pencil 
 of tangents is moved to the point at the extremity of the force 
 vector the points of intersection of the tangents with correspond- 
 ing acceleration vectors form a circle which is the limiting form of 
 the quartic of Theorems IV, V and VI, and whose diameter is 
 the length of the force vector. 
 
 Having found the length of the force vector its projections 
 on the tangent and normal for any lineal element as well as the 
 projections of the space derivative of the force vector can be found. 
 
 7. The radius of curvature of the conic of Property 1. 
 
 The equation of the conic of Property 1 belonging to the 
 general system (32) is 
 
 (36) 3 y" 3 (3 y"A - 5) X-+3 v" 2 B X (Y-y'X) 
 
 + C (Y-y X)--3 /" (Y-y X)=o 
 
 (Kasner, Bulletin Amer. Math. Soc, March, 1907.) 
 In the present problem 
 
 07) A = 5 ^+y'» 
 
 B=- 1 
 
 D 
 
 -3 OA-vV) y' n 
 
 +(io/p 1 +)p 2 (i+/ 2 )y 
 
 + (1+/ 2 ) 2 P 3 
 
 c = 
 
 D{\+y'*) 
 
 9^y" 4 +3 j(5y' 2 - 1) Pi+y'Pz] y" 3 
 
 1 P 
 
 £(l+/ 2 )[ +3 y'/Ml +/' 2 ) /' 2 
 where Pu P>>, Ps, D represent the values assigned them in (30). 
 
CONVERSION OF PROPERTIES. 23 
 
 Substituting (37) in (36) we have the conic which is the locus 
 of centers of osculating conies for curves having a given curvature 
 element in 5. 
 
 (38) A' 2 [3 tW 2 +(5-/ 2 ) P iy "-P2y'y"-P s y' (l +y' 2 )] 
 
 +x y [-3 a-/?) y /2 +i2 p 1 yy / +p 2 y'(i-y 2 ) 
 
 + P 3 (l-y' 4 )] 
 
 + V 2 [3 ^ y" 2 + { (5 y' 2 - 1) Pi+y'P 2 ] y"+Pi/ (i+/ 2 )] 
 
 + X D y' (1+/ 2 ) -YD ( 1 +y"-) =0. 
 
 Differentiating (38) with respect to X we find 
 
 , , 10y"P,(l+/ 2 ) 
 
 r y , ^ 
 
 The radius of curvature for conic (38) is then 
 
 (39) dV 1+3/2 Vi+y 
 
 10 /'A 1 
 
 _ j_ f 3 (y+yv) vi+7 2 d+y 2 ) 3 H j_ r» J 
 
 ~To[_ Pi y" J 10 L j 
 
 This is Property 3 and is expressed by 
 
 Theorem XL The radius of curvature of the conic corre- 
 sponding to a given curvature element by Theorem IX is one-tenth 
 the difference of the radius of curvature for those curves having 
 third order contact with their circles of curvature minus the radius 
 of curvature of the given curvature element. 
 
 8. The locus of the second center of curvature of the conic 
 of Property 1 is a certain cubic. 
 
 The general formulas for the second center of curvature of 
 a given curve are 
 
 (Mis r - -4y'(l+;/ 2 ) /"(!+/') 
 (4 °) *2 = y, 1 yr 3 ; 
 
 (1+y 2 ) (1-3 / 2 ) , yV"(l+y' 2 ) 2 
 
 y* = j, + —jh • 
 
 For the conic of Property 1, Y'=y', 
 10 V'P, (1+/ 2 ) 
 
 y= 
 
 D 
 
 „,_ 30 y"Pi [ -3 U-y'v) y" 2 +(l+y' 2 ) (10 y'P x +P 2 ) y'H 
 D* +(l+y' 2 ) 2 P 3 . J 
 
24 CURVES OF CONSTANT PRESSURE. 
 
 Substituting these derivatives for y', y", y'" in (40), 
 
 [- 10 y'y" (1+/2) Pl Z) + 3 D [_ 3 ty-ftp) /"] 
 
 (AU ,_ L -p 2 (i+/ 2 )y r + (i+y) 8 P«i • J 
 
 K * l) x ~ ioo y 2 (i+/ 2 ) Px 2 
 
 ^ 2 : 
 
 fio y (i+y) Pi £+3 yz? [-3 (tA-yV) y 2 ] 
 
 L +p 2 d+y 2 ) y+ a+y /8 )p»i J 
 
 ioo y 2 (i4-/ 2 ) Pi 5 
 
 To find the locus of the second center of curvature of conic (38) 
 as y" varies, keeping y' constant, y" must be eliminated from the 
 above equations (41). The elimination gives the following result, 
 when the equation is simplified. 
 
 (42) 3 P 3 (}'2~y'x 2 ) [3 (1+/ 2 ) (*+yV)-ro Pi (>-2-A 2 )] 2 
 
 4- Pi (1+/ 2 ) [3 (1+y 2 ) (^4-/^-10 Pi (r 2 -A 2 )] 
   [- 10 Pi 6> 2 /+* 2 )+3 P 2 (y 2 -y'x 2 )} 
 
 -9Pi 2 (i+y 2 ) («£-y>) (^ 2 -y 2 )=o. 
 
 The general character of this curve is more easily seen by writing 
 the equation in the form when y' has the special value zero. 
 
 (43) 3 P 3 y 2 (10 Pij 2 -3 <p) 2 
 
 + Pi (10 P& -Zip) (10 Pix 2 - 3P 2 y 2 ) 
 -9 P^y 2 = o. 
 
 The curve is a cubic passing through the origin once and having an 
 asymptote parallel to the direction of the element y' which passes 
 through a double point at infinity. Every line parallel to the 
 element thus cuts the curve in only one finite point. Property 4 
 may now be stated as 
 
 Theorem XII. As the curvature of a given element varies, 
 leaving the direction and point fixed, the second center of curva- 
 ture of the conic of Theorem VIII describes a cubic curve which 
 passes through the fixed point once and has an asymptote parallel 
 to the element with a double point on the asymptote at infinity. 
 
 9. The intercept of the asymptote of the cubic on the normal. 
 
 Directly from the equation (42) we see that the equation of the 
 asymptote is 3(1 +y'-) (^+/^)-10 Pi {y' 1 -y'x 1 )=Q and its 
 
 intercept on the r-axis is 3 - ^ — Tr±. The projection of 
 
CONVERSION OF PROPERTIES. 25 
 
 .3 (14-/ 2 ) v+y'ip 
 
 this intercept on the normal gives — - — ^ — . , and this 
 
 6 10 Pi V1+/2 
 
 is the intercept of the asymptote on the normal since the asymptote 
 
 is perpendicular to the normal. The intercept may be expressed 
 
 3 T 
 in terms of known vectors — - v? an d furnishes Propertv 5 for 5. 
 
 10 AT F - 
 
 Theorem XIII. The intercept of the asymptote of the cubic 
 of Theorem XII on the normal to the element to which the cubic 
 belongs is three-tenths the ratio of the tangential component of the 
 force to the normal component of the space derivative of the 
 force. 
 
 10. The direction of the cubic at the origin. 
 
 The slope of the cubic (42) at the origin is 
 
 [9 p,/( v? +/^)2+ 10^2(^4-/^)4-3 P,P 2 /(^4-j-V) 1 
 
 , L -3 Pry (iA-.-vV) J 
 
 y * " \9 p 3 (^4- J v) 2 -io PiVO+yW+3 PiP2(<p+y'+) 1 
 
 L -3 PS (+-y'<p) J 
 
 whence the tangent of the angle which this direction makes with 
 the normal to the chosen element is 
 
 <«> -3[3P,(^) + ^-^^} 
 
 10 Pi 2 
 
 In order to obtain a geometric interpretation for this expression 
 it is necessary to write down the components of the second space 
 derivative of the force vector. 
 
 The normal component is found to be 
 
 (14-/ 2 F 
 
 (i+/»r 
 
 For those curves which have third order contact with their circles 
 
 * + t. r/ Pi(l+v' 2 ) 
 
 of curvature, when V = „ / , ' , ' , 
 
 3 (<P+y W 
 
 ft = ft 3P,(y+yV)+P# l ,-y(y l> +^)+3r f V,] 
 
 3 (*+•*) (14-/^ 
 
 Now P 2 = *„(l-4 / 2 )-f Vx(/ 2 -4)~S /{**+**) 
 
26 
 
 CURVES OF CONSTANT PRESSURE. 
 
 .-. 3 (*>+/*) P 3 +PiP2 = N c 3 (?+/*) (l+y 2 )' : 
 
 =AT e 3 7(l+y 2 ) 2 -4AT • T (1+/ 2 ) 2 
 (44) becomes 
 
 _3_ 
 
 10 
 
 To 
 
 3 JV e r-4 iV T+N 2 ^^ / } 
 
 N 2 
 
 3N C T AT 
 
 W iv 
 
 
 l 
 
 
 This is Property 6 and may be expressed in words as 
 
 Theorem XIV. The tangent of the angle which the cubic of 
 Theorem XII makes with the normal to the element to which the 
 cubic corresponds is three-tenths the sum of three ratios. The first 
 ratio is negative three times the product of the normal component 
 of the second space derivative of the force, as specialized for those 
 curves which have third order contact with their circles of curva- 
 ture, multiplied by the tangential component of the force divided 
 by the square of the normal component of the first space derivative 
 of the force. The second ratio is four times the tangent of the 
 angle which the first space derivative of the force makes with the 
 normal. The third is the negative tangent of the angle which the 
 fixed element makes with the force vector. 
 
 (45) 
 
 1 1 . Hyperosculating parabolas. 
 
 The differential equation for all parabolas in a plane is 
 5 y'" 2 
 
 y'2 = 
 
 3/ 
 
 Those curves of S which have contact of the 
 
 fourth order with parabolas must have the same value of y iv . 
 Substituting (45) in (31) the equation is seen to be quadratic in 
 y'" . Hence of the oo 1 curves of (31) having a given curvature 
 element two have contact of the fourth order with their osculating 
 parabolas. The locus of the foci of these hyperosculating parabolas 
 as y" varies will now be found. The coordinates of the focus of 
 a parabola in terms of derivatives are 
 
(46) 
 
 z a 
 
 2(3- 
 
 CONVERSION OF PROPERTIES. 27 
 
 -3 y »[y'"(y'«—\)+2 y'(3 y' 2 -//")] 
 
 y'"2+(3 y"*-y'y'"y 
 
 -3 y"[(3 y'" l -y'y"') (/ 2 -l)-2 //"] 
 
 }'"' 2 4-(3 y"*-y'y'"y 
 
 Solving these equations for y" and y'" 
 (14-/2) (£-«/) 
 
 (47) /' = 
 
 2 (a 2 4-/3 2 ) 
 
 „, = 3 (1+/ 2 ) (/3-a /) [a+/3 y' + y' (/3-a y')) 
 y 4 (a 2 4-/3 2 ) 2 
 
 When /', 7"' and y IV from (47) and (45) are substituted in (31) 
 the locus of the focus (a, /3) as y[' varies will be found, a and /3 
 being referred to the chosen point (x, y) as origin. The resulting 
 equation simplified is 
 
 (48) 4 P 3 (a 2 4-/3 2 ) 2 (« + /3 y') - 10 Pi(a 2 4-/3 2 ) (a+/3 y') 2 
 
 4-2 Pifa s +/3«) (/3-a y') 2 + 2 P 2 (a 2 + /3 2 ) (a+0 V) (/3-a/) 
 -3 ty-y<p) (/3-a y) 2 (a +/3 /) 
 -3 (*>+/$ (0-a y'y = o 
 
 representing a curve of the fifth degree having a triple point at the 
 origin and the element as a double tangent. 
 
 Theorem XV. Of the °° ' curves of 5 having a given curvature 
 element two have fourth order contact with their osculating para- 
 bolas. The locus of the foci of the hyperosculating parabolas as 
 the curvature varies, leaving the lineal element fixed, is a quintic 
 curve having a triple point at the origin and having the element 
 tangent as a double tangent. 
 
 The slope of the third branch at the origin is found to be 
 
 — , . , , . , — ^-;-\ and the angle which this line makes with the 
 
 v+y't+y'W-yv) 
 
 element is the same as that which the element makes with the 
 line of force. 
 
 Theorem XVI. The single branch of the quintic of Theorem 
 XV at the fixed point is so situated that its tangent at the fixed 
 point makes with the line of force an angle bisected by the element. 
 
 Theorems XV and XVI express Property 7. 
 
28 CURVES OF CONSTANT PRESSURE. 
 
 12. The point of intersection of the quintic with the chosen 
 element. 
 
 Let jS — ay' = o. We find then 
 
 5 P 
 (49) a = — p 1 and 
 
 P 2P 3 
 
 as the coordinates of the only point of intersection of the element 
 tangent with the quintic (48) in addition to the points at the origin. 
 The distance from the origin to this single point Q of intersection is 
 
 5 P x Vl+V 2 = 5iV_ 
 2 P 3 " 2 Ni ' 
 
 In seeking the geometric explanation of this quantity it is necessary 
 to observe that W.-ff- ^'"^^*^ 1 . 
 
 (1+? ) ' 
 
 Now since the quintic of Theorem XVI is the locus of foci of oscu- 
 lating parabolas the point Q is the focus of a particular parabola. 
 Using again the general coordinates of the focus (46) and giving 
 them the special values (49) it is found that the only curvature 
 possible in this case is given by y" — o. Hence we may say that 
 
 N\ = N tt where N a indicates the normal component of the second 
 space derivative of the force for those curves which are hyperos- 
 culated by the particular parabola whose focus is Q. We now 
 state as Property 8 the facts just proved. 
 
 Theorem XVII. The quintic corresponding to a given lineal 
 element by Theorem XV intersects the element tangent at one 
 point besides the point of the element. The distance of this 
 point from the point of the element is five-halves the ratio of 
 the normal component of the first space derivative of the force 
 to the normal component of the second space derivative of the 
 force for those curves of constant pressure which are hyperosculated 
 by the particular parabola whose focus is thus defined by the inter- 
 section of the quintic and the element tangent. 
 
CHAPTER IV 
 
 The Converse Problem for 5. 
 
 1. Converse of Property 1. 
 
 Professor Kasner has proved that Property 1 is characteristic 
 of all sets of °o 4 curves whose equations are of the form 
 (32) y iy =A y'"-+B y'"+C, where A, B and C are arbitrary 
 functions of x, y, y', y" . It will now be shown that the other 
 properties already stated are sufficient to specialize the equation 
 (32) until it takes the form (31) which is known to represent all 
 the curves of constant pressure in a plane field of force. 
 
 2. Converse of Property 2. 
 
 Since the circles of a plane are included in the system (32) the 
 differential equation of the circles must satisfy the equation (32). 
 
 Substituting V" = ^C, -v IV = V'1 ( ^ 2 C +1) in (32) we have 
 
 1+y- (1+/*) 
 
 (50) 3/' 3 (5/ 2 +l)=VyM4+3/y /2 (l+/ 2 ) B+{\+y'"YC, 
 
 a relation among A, B and C in consequence of Property 2. Solv- 
 ing for C and substituting the resulting value in (32) we have the 
 equation of all curves possessing Properties 1 and 2. 
 
 (51) f*=A y'" 2 + B y'" 
 
 . $ y" z {5 y' 2 +\)-9 y'ytA-Sy'y"* (1+/ 2 ) B 
 
 (1+/ 2 ) 2 
 
 C will still be written instead of the long final term but it is under- 
 stood that it may at any time be replaced by its value in terms of 
 A and B as A and B are modified by later properties. 
 
 3. Converse of Property 3. 
 
 In Property 3 we must make use of all the information implied 
 in the expression "radius of curvature of those curves which have 
 third order contact with their circles of curvature." In the 
 converse problem the force vector and its derivatives are to be 
 treated as purely geometrical vectors. The force vector will be 
 
 called F and its first and second space derivatives F and F respect- 
 ively. The projections of F on the tangent and normal for any 
 given element will be called T and N and the corresponding pro- 
 jections of F and F will be T, N, T and N. Expressed without 
 assuming the existence of a field of force, the information involved 
 in Property 3 is then as follows : 
 
 (29) 
 
30 CURVES OF CONSTANT PRESSURE. 
 
 For each point in the plane there exists a vector F determined 
 
 in direction and magnitude by the curves of 5 passing through the 
 
 point which have third order contact with their circles of curvature. 
 
 For every lineal element at the point there is one circle of curvature 
 
 which has four points in common with each of oo 1 curves of the 
 
 system 5. As the direction of the element varies the center of this 
 
 circle describes a conic which passes through the fixed point in a 
 
 certain direction indicated hereafter by y' = <a, the direction of the 
 
 vector F. The hyperosculating circle itself is a member of the 
 
 system 5. The set of all circles in the plane, called 5«, forms a 
 
 distinct part of 5 such that when the corresponding tangents to 
 
 focal circles are considered according to Property 3 of the third 
 
 order problem a circle appears as the limit of the quartic in 
 
 Property 3. The diameter of this limiting circle is the length of 
 
 the vector F. The radius R c of the hyperosculating circle for any 
 
 3 T 
 element (x, y, y') is then geometrically expressed by — =. The 
 
 radius of curvature of the conic of Property 1, called briefly 
 conic I, is one-tenth the difference of R c minus the radius of 
 curvature for the curvature element to which conic I corresponds. 
 When the vector F is referred to any pair of rectangular axes 
 through the fixed point its projections on the axes may be called 
 
 >p and \p and its direction w = -. <p and \p are thus functions of x and 
 
 <P 
 
 y and determine a field of force. The length of F may now be 
 
 expressed V \p 2 -\-\J/' 2 and the projections of F on any tangent whose 
 direction is y' and the corresponding normal are 
 
 Also the components of the space derivative F of F along the axes 
 are <p 8 and ^ 8 and the components of F along the tangent and normal 
 
 Hence the radius of curvature for conic I is known geometrically 
 from Property 3 in terms of known vectors. 
 
CONVERSION OF PROPERTIES. 
 
 31 
 
 The radius is 
 
 To 
 
 3(^4-/*) (14-/ 2 ) 3 / 2 
 
   y ^ s 
 
 Vi+/ 2 
 10 
 
 3 /'(«*+/*)- 
 
 V)-Pi(l +/') "! 
 
 The general conic (36) corresponding to the typical fourth order 
 
 equation (32) has a radius of curvature 
 
 (1+/ 2 ) 3 -' 2 
 
 If the 
 
 2(3 A y"-5) y"' 
 curves of 5 have Property 3 this radius equals that given above. 
 
 (1+/ 2 ) 3 
 
 V 
 
 2 (3.4 y"-5) y" 
 Solving for A we have 
 (52) .4 = 
 
 l + v /2 ["3v" 
 10 "[ 
 
 (v+y'tf-Pxil+y*) 
 
 Piy" 
 
 5 {<p+y'<p) 
 
 3?'{iP+y'*)-Pi(\+y Ji ) 
 
 Hence a set of °o 4 curves having Properties 1, 2, 3 are curves 
 in a field of force and are represented by the equation 
 
 (53) ^ = 
 
 5 (v+y'v) 
 
 v 
 
 ' + B y'"+C 
 
 3y"(<P+y'f)-Pi(i+y' 2 ) 
 
 where P] is the known function of x, y, y' given in (30), B is an 
 arbitrary function of ,x, y, y', y" and C is related to B according 
 to (50). 
 
 The relation (50) may now be modified by using (52) to elim- 
 inate A. 
 
 4. Converse of Property 4. 
 
 The most general equation of a cubic which passes through 
 the origin once, has an asymptote parallel to the *-axis and has 
 a double point on the asymptote at infinity is the following : 
 
 y*+A, * (y-4o)+4 3 y 2 +4 4 y = o. 
 
 A corresponding equation through any point (v, y) as origin and 
 having the asymptote parallel to the direction y' is 
 
 (54) (y 2 -y.'V2)' ! +5 1 (,v,+-y'y 2 ) (y 2 -y'.v 2 - B 2 ) 
 
 4- B 3 (y 2 - y%) *+B 4 (y 2 - y'x 2 ) = o 
 
 where ;v 2 , y* are referred to the fixed point (x, y) as origin, y' is any 
 fixed direction and Pi, P 2 , P.s, B A are arbitrary functions of x, y, y'. 
 
32 CURVES OF CONSTANT PRESSURE. 
 
 Property 4 requires that such a cubic shall be the locus of the 
 second center of curvature for conic I as the direction of the 
 element y' varies. According to formulas (40) the coordinates of 
 the second center of curvature for conic (36) are 
 
 .... -D(40yy'P,+3BZ)) 
 (55) Xi = 
 
 yi = 
 
 100 y" 2 Pi 2 
 D f 10 ZV'(1 -3 y' 2 ) -3BD y'\ 
 
 100 y" 2 Pi 2 
 from which we get 
 
 d (1+y 2 ) 
 
 y'2—yx 2 = 
 
 x 2 +y'y 2 = 
 
 10 y"P x 
 
 -3D (14-/ 2 ) (10 Piy'y"+B D) 
 
 100 y" 2 Pi 2 
 
 Substituting these values in equation (54) and simplifying, 
 D 2 (14-/ 2 ) 2 -3 BxD (1+/ 2 ) (10 Piy'y"+B D) 
 4-30 BxB 2 Pxy"(lO P iy 'y"+B D) 
 4-10 £ 8 (1+/ 2 ) Piy"D+l00 BW' 2 PS = o. 
 
 From this B can be expressed in terms of B u B 2 , B 3 , B. u 
 
 By substituting for B in (53) we obtain the equation of a set of 
 
 curves having Properties 1, 2, 3, 4. 
 
 (56) y"=. 5 (*+3W ?"" 
 
 j 2! I z) 2 (i 4-/ 2 ) 2 
 
 ^3 B X D [D (1+/2) _ io p^o/'jL v ^ y 
 
 -30 ^Z) Pi/y'(l+/ 2 ) 4-300 BxB.Px^y'y'" 1 
 
 4-10 5 S P,D /'(1 4-/2) 4- 100 £ 4 /' 2 Pi 2 l 
 
 4-C. 
 
 In this form Pi and D are defined by (30), B lt B 2 , B 3 , £ 4 are 
 undetermined functions of x, y, y' and C is related to the other 
 quantities by relation (50) which may now be rewritten sub- 
 stituting the value of B. 
 
 5. Converse of Property 5. 
 
 As the general cubic (54) is written it is evident that B 2 
 represents the intercept of the asvmptote on the y 2 -axis. Its 
 
 B 2 
 
 intercept on the normal to the direction y' is then 
 
 Vl+/ 2 
 
CONVERSION OF PROPERTIES. 33 
 
 According to Property 5 this intercept must be three-tenths the 
 ratio of the tangential component of the force to the normal com : 
 ponent of the space derivative of the force. Hence 
 
 (57) - B 2 = ^ . 
 
 Substituting this value of B 2 in (56), the equation of curves having 
 Properties 1, 2, 3, 4, 5 is 
 
 (58) y 
 
 5 (y+yV) /"' 
 3 y "(„+/f)_p 1 (i+/*) 
 
 v'" 
 
 +30 £ 3 Pi(l+;/ 2 ) (^+rV) + 100 £ 4 Prj 
 4-/'[-6Pi(l+/ 2 ) 3 (^+yV)+30SiP 1 y(14-y2)2 
 
 -io^PxMi+Z 2 ) 2 ] 
 
 +Pl 2 (l+;/ 2 ) 4 } 
 +c, 
 
 where the only unknown coefficients are B\, B 9 , B A functions 
 of x, y, y'. 
 
 The relation (50) of C to B is further specialized by the new 
 value of B. 
 
 6. Outline of the final steps of proof. 
 
 It remains to show that the coefficients B\, B 3 , B A are deter- 
 mined by Properties 6, 7, 8 so that the equation (32) is finally 
 reduced to the form (31) representing curves of constant pressure 
 in the field of force (<p, \J/) . It will be seen that B A and B 3 are found 
 in terms of B x and finally Bi in terms of known quantities. After 
 each step a substitution can be made in (58) eliminating one of 
 the unknown coefficients but the resulting expressions are long and 
 the actual writing of the modified equation will be postponed until 
 the characterization of the curves of 5 is complete. 
 
 7. Converse of Property 6. 
 
 In order to impose the requirement of Property 6 upon the 
 set of curves represented by (58) the cubic (54) at the point of 
 the element to which it belongs must make with the normal to the 
 element the angle described in Theorem XIV. The tangent to 
 the cubic (54) at the fixed point is found to be 
 y 2 B&Ary'B* 
 
 x 2 -B x B«y'A-B; 
 
 The tangent of the angle which this line makes 
 
34 CURVES OF CONSTANT PRESSURE. 
 
 D 
 
 with the normal is —5-^-. Replacing B t bv its value from (57) 
 
 and setting this expression equal to the tangent of the angle 
 described in Theorem XIV, we find B± in terms of B x by solving the 
 resulting equation 
 
 100 P, 3 [ -P^^-^V) J- 
 
 The substitution of (59) in (58) produces the equation of 
 curves having Properties 1, 2, 3, 4, 5, 6. Call that equation (60). 
 
 8. Converse of Property 7. 
 
 Following the method employed in Article 10, Chapter III, it 
 is found that two curves of equation (60) having a given curvature 
 element have contact of the fourth order with their osculating 
 parabolas. The locus of the foci of the hyperosculating parabolas is 
 
 (61) 4 P 1 2(l4->^)2( a 24-/3 2 )2(«4-/3 y') 
 
 -30 ^P^l+y' 2 ) (a 2 4-£ 2 ) (a+By'V 
 4-6P, 2 £ 1 (14-y' 2 ) (a 2 4-/3 2 ) 03-a/) 2 
 -4P,(l + y' 2 ) [3 (1+y 2 ) (^4-vV)+5 £ 3 Pi] (a 2 +£ 2 ) (jS-a/) 
 
 («+/3 /) 
 4- [9 (14-/ 2 ) 2 (^+yV) 2 +30 ^Pjd+y' 2 ) (*+yty) 
 4-100 £ 4 Pi 2 ] ()8-a y') 2 (a+(3 3;') 
 
 -9 PiE^l+y 2 ) (<^4-yV) (<3-ay') 3 = o, where £ 4 is written 
 for brevity instead of (59). The form of this equation shows that 
 every set of 00 4 curves having Properties 1,2,3, 4, 5, 6 has also the 
 property that the foci of the hyperosculating parabolas correspond- 
 ing to a given lineal element lie on a quintic curve which has a 
 triple point at the origin and has the element tangent as a double 
 tangent. 
 
 Consider now the equation of the tangent to the third branch 
 of (61) at the origin 
 
 (62) J -9 (14-y 2 ) 2 (^ + jV') 2 -30 B«Pi{l+/«) (^4-yV) 
 £_ 1 -100 B 4 P{--9 ff 1 P 1 y'(l+;/ 2 ) (y+yY) 
 
 a / 9 /(l+/ 2 ) 2 (^+/,A) 2 +30 # 3 P,(1+/ 2 ) /(*>+/*) 
 I 4-100 B i Pfy'-9 BiPrd+y' 2 ) (*>+/£) 
 
 The tangent of the angle which this line makes with the element 
 y'is 
 
 (63) 9(14-/ 2 ) 2 (^4-yV) 2 +30^Pi(l+y /2 ) (^4-/^)4-100 B 4 P X 2 
 
 9£ 1 Pi(14-ri 2 ) (y+yV) 
 
CONVERSION OF PROPERTIES. 35 
 
 To satisfy Property 7 this must equal the tangent of the angle 
 
 § j 
 
 which the element makes with the line of force, — ' .. . 
 
 When this equation is formed and solved for B 3 , using expres- 
 sion (59) for B 4 , we have 
 
 (64) B a = - 
 
 10 Pi 2 
 
 Pi(l+/ 2 ) (?+vty)+3 5,P,(^+vty) 
 +B 1 P 1 P 2 
 
 Substituting this in the fourth order equation (60) as already 
 modified by Properties 1, 2, 3. 4, 5, 6 we shall have (65) as the 
 equation of curves having all seven properties so far employed in 
 the converse problem. 
 
 9. Converse of Property 8. 
 
 The point of intersection of the quintic (61) with the element 
 
 y' is found by substituting 8 — a y' — o in equation (61). The co- 
 
 r .1 • 15 Pi _, 15 Biy' 
 
 ordmates of the point are a = ^ ,, — rr. and 8 = - ,, , ,„. and 
 
 2 (1+V 2 ) 2(1+/ 2 ) 
 
 its distance from the fixed point of the element V a 2 +/3 2 = - , -. 
 
 2Vl+/ 2 
 
 By Property 8 this distance must be five-halves the ratio of the 
 normal component of the vector F to the normal component of the 
 
 vector F for those curves which are hyperosculated by the par- 
 ticular parabola whose focus is the intersection of the quintic and 
 the element tangent. Expressed in symbols this property gives 
 the equation, 
 
 15 B l 5 P, Vl-f-/ 2 
 
 -^— . Hence 
 
 2 Vl-f/ 2 2 P 3 
 
 Pi(l + / 2 ) 
 
 (66) Pi 
 
 2>Pz 
 
 Now substituting (66) in (65) and simplifying, the fourth order 
 equation takes the form of equation (31) representing all the 
 curves of constant pressure in the field (<p, \p). 
 
 It has now been proved that the eight properties derived in 
 Chapter III are sufficient to identify a set of oo 4 curves in a plane 
 as curves of constant pressure in a field of force. 
 
VITA 
 
 Eugenie M. Morenus, born in Cleveland, New York, February 21, 
 
 1881. 
 Bachelor of Arts, Vassar College, 1904. 
 Master of Arts, Vassar College, 1905. 
 Graduate Student, University of Chicago, University of Gottingen, 
 
 Columbia University. 
 Teacher in Watertown High School, January, 1906, to June, 1907. 
 Substitute in Mathematics, Vassar, 1907-1908. 
 Teacher in Poughkeepsie High School, 1908-1909. 
 Instructor in Mathematics and Latin, Sweet Briar College, 1909- 
 
 1916. 
 Associate Professor of Mathematics, Sweet Briar College, 1916- 
 
 1918. 
 
 Professor of Mathematics, Sweet Briar College, 1918 
 
 Master's Thesis: "Some Curves Connected with a System of 
 Similar Conies." 
 
r. 
 
478713 
 
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