PROPOSITION 90. Problem.

821. For a given path Mmμ on some surface (Fig. 91) to find the position of this path with respect to a given plane APQ, and of the radius of osculation of this path at M, as long as neither the position nor the length of the radius lies on the surface.

Solution.

With the plane APQ taken for argument’s sake [in the plane of the page] and in that plane the axis AP is taken, with respect to which the position of the curve Mmμ is to be determined ; now from three nearby points M, m and μ of the given path on the surface the perpendiculars MQ, mq, μρ are sent to the plane APQ and the perpendiculars QP, qp, and ρπ [are dropped] from the points Q, q, ρ to the axis AP. Now the initial position of the abscissa at A are AP = x, PQ = y and QM = z. Again since the given surface is put in place, an equation is given expressing the nature of this between these three variables x, y and z; and this equation is [of the form]: dz = Pdx + Qdy . Since if this equation is connected with another, a certain line present on the surface is expressed; whereby, as the given line Mmμ is put in place, as well as the equation dz = Pdx + Qdy above, another equation is given, from which the curve Mmμ can be determined, [p. 458] but there is no need to represent that here. Let the elements of the abscissa Pp , pπ = dx be equal to each other, or the element is dx taken to be constant. [The derivations that follow rely heavily on the section §68 onwards at the end of Ch. 1] Hence there is: and With these in place, let MN be the normal to the surface at the point M, and N the point at which this normal crosses the plane APQ ; the perpendicular NH is sent from N to the axis ; then (68). ∂z and Q = ∂z at M. [For we can write in modern terms : dz = ∂∂xz dx + ∂∂yz dy ; hence P = dx ∂y If the line QH’ (not shown) is drawn parallel to AH in the xz-plane, we have QH ' = tan(QMH ’ ) = ∂∂xz , and hence we have the subnormal QH’ = z∂∂xz = Pz as required. MQ Similarly, for the line QN’ (not shown) is drawn parallel to HN in the yz-plane, we have QN ' = tan(QN ’ M ) = ∂∂yz , and hence we have the subnormal QN’ = z∂∂yz = Qz as required, MQ and the signs can be taken into account ; see Euler’s explanation and the note on page 19. Note also that Euler has in mind very simple surfaces such as those of cylinders, cones, and surfaces of revolution about an axis, so that only one radius of curvature has to be found. You may wish to copy the above figure and annotate it, as this helps greatly in understanding the working.] Now let MR be the position of the line of the radius of osculation of the curve Mmμ and R the point of incidence of this in the plane APQ; then with the perpendicular RX sent from R to the axis : andEULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 695 (68). Now the length of the radius of osculation , clearly MO, is equal to (72). Finally the plane considered, in which the elements Mm , mμ are in place, is produced until it intersects the plane APQ, and let the line of intersection be RKI, which the perpendiculars from A and P cross at K and V; it was found above (68) that Now since we have XR − PV : AX − AP = PV : PI , then Now and With which in place, we have : [p. 459] and Hence, it is found that : Now the inclination of the plane in which the elements Mm et mμ are placed to the plane APR can be found by sending the perpendicular QS from Q to the line of intersection RI; QM for the tangent of the angle of inclination is equal to QS . But since IV : PI = QV : QS , then that tangent is equal to :EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 696 Now the tangent of the angle NMR, that the radius of osculation makes with the normal to the surface, is equal to (71) : Therefore from these everything can be deduced that is required in understanding the position of the curve Mmμ . Q.E.I. Corollary 1. 822. The projection of the curve Mmμ in the plane APQ is the curve Qqρ , the nature of which is expressed from the equation between x and y. Whereby this projection is obtained, if with the help of the equations dz = Pdz + Qdy and that by which the curve is determined on the surface, a new equation is formed from the elimination of the variable z, which is between x and y only. Corollary 2. 823. In a like manner, if x is eliminated, in order that an equation is produced between y and z, from this equation the projection of the curve Mmμ is defined in the plane normal to the axis AX. [p. 460] And the equation, in which y is not present, but only x and z, gives the projection of the curve Mmμ in the plane normal to the plane APQ cutting the axis AX. Corollary 3. 824. But the nature of the curve Mmμ is known distinctly from any two of these normal projections in two of the planes in turn. Such knowledge is also supplied by a single projection together with the surface itself. Corollary 4. 825. On account of which the curve on the surface is required to be designated by some characters as well as the equation dz = Pdx + Qdy , from which surface is determined, and an equation is given involving only two variables for some projection of the curve Mmμ . Corollary 5. 826. If the surface is cut by a plane, in a like manner to that in which the cone is accustomed to be cut producing the conic sections, then the curve arises from this section is in the same plane. Whereby in these cases as the position of the right line IR is constant so the inclination of the plane IMR to the plane APQ. Example. 827. Therefore if some surface is given and that is cut by the plane IMR, the curve is sought arising from this section. [p. 461] Accordingly, there is put in placeEULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 697 AI = a , AK = b and the tangent of the angle of inclination of the plane IMR to the plane APQ is equal to m; then zdxddy − ydxddz a = dzddy − dyddz − x and b = zdxddy − ydxddz − xdzddy + xdyddz dxddz and m= ( dx 2 ddz 2 + ( dzddy − dyddz ) 2 ) dxddy From which equations joined with dz = Pdx + Qdy the nature of the curve generated by this section can be determined. Now from previously from the two equations there arises b = dzddy − dyddz or ddz : ddy = adz : bdx + ady ; a dxddz and the integral of this equation is : 1 ldz = 1 l (bdx + ady ) - 1 lc or cdz = bdx + ady a a a and again Now in the first equation if in place of ddz and ddy the proportionals of these are substituted, there is produced a+x= bzdx + azdy − aydz or abdz + bxdz = bzdx + azdy − aydz , bdz and the integral of this divided by zz is this : c − ab

z

bx + ay

or cz = bx + ay + ab;

z

hence what before was ff, this is ab, or ff = ab. Now the constant c of the third equation

can be defined ; moreover then

dz ( a 2 + b 2 )

m = bdx + ady

or

dz ( a 2 + b 2 )

= bdx + ady .

m

Where the above letter is

c=

(a 2 +b 2 )

m

and in addition the nature of the surface is expressed by this equation :

z (a 2 +b 2 )

= bx + ay + ab ,

mEULER’S MECHANICA VOL. 2.

Chapter 4a.

Translated and annotated by Ian Bruce.

page 698

from which the nature of the curve sought can be derived. Moreover because the whole

curve sought is in the plane IMR, most conveniently that can be expressed from the

equation between the orthogonal coordinates taken in the same plane. Hence with IR

taken for the axis, from M to that there is sent the perpendicular MS and calling IS = t and

MS = u. Now we have IA : AK = IP : PV or

and

Again we have [p. 462]

(a 2 + b 2 ) : a =

z (a 2 +b 2 )

QS ; ma whereby From these there is produced : From which there arises : and with these values substituted in the equation z ( a 2 +b 2 ) = bx + ay + ab there is m produced : Therefore with these values substituted in place of x, y and z in the equation of the surface there comes about the equation between t and u, or the orthogonal coordinates of the curve sought. Corollary 6.

828. If the intersection of the cutting plane IR falls on the axis AX and I is taken at A, thenEULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 699 Corollary 7.
829. If the intersection IR of the cutting plane IMR with the plane APQ is normal to the axis AX, then b =∝ . Whereby there is produced : Corollary 8.
830. Since the values to be substituted in place of z, y and x are of one dimension of t and u, it is evident that the equation between t and u is not possible to have more dimensions than the equation itself between z, y and x. Corollary 9. [p. 463]
831. Whereby if the equation between z, y and x is of two dimensions, there are many surfaces of this kind given in addition to the cone, all the sections made by a plane are conic sections. Scholium.
832. In that dissertation in Book III of the Commentaries [of the St. Petersburg Ac. of Sc.], in which I have determined the shortest line on a surface, I have pursued three kinds of surfaces, which are the cylinder, the cone, and the surface of revolution. [L. Euleri Commentatio 9 (E09): Concerning the shortest line on a surface joining any two points. Comment. acad. sc. Petrop. 3 (1728), 1732, p. 110; Opera Omnia series I, vol. 25.] Now the general equation dz = Pdx + Qdy gives a cylindrical surface, if P vanishes and Q depends only on y and z, thus so that the abscissa x does not enter the equation for this kind of surface ; for all the sections are parallel to each other and are equal also ; for these the equation is therefore dz = Qdy . I refer all these surfaces to the genus of confides, which are generated by drawing right lines from some points of an individual curve to a fixed point placed beyond the plane of that curve. Which surfaces have this property, that all parallel sections are similar to each other and the homologous lengths of these are as the distance of the sections from the vertex of the cone. Now equations for the surfaces of this kind, if indeed the vertex of the cone is at A, thus are compared, so that x, y and z everywhere together constitute a number of the same dimensions. Finally I have turned or rounded surfaces [of revolution], which are generated by the rotation of any curve about an axis ; if AX were such an axis, on putting x constant, the equation between y and z gives a circle with centre P. Whereby the equation for these has this form : [p. 464]EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 700 y where Pz only depends on x ; or Q = − z and P = Xz with X present as a function of x. Moreover as in these turned surfaces all the sections are circles normal to the axis, thus such surfaces can be taken, the sections of which are any similar curves normal to the axis. All such surfaces hold this general property, that any function of x is everywhere equal to a function of y and z of the same number of dimensions. As, if the number of this dimension is n, for this is a property of the equation Pdx = Rdz + Qdy that it is [See E044.] From which, or the equation for a surface of this kind can at once be concluded from what has been given.EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 701 PROPOSITION 91. Problem.
833. In any given surface to determine the line, that the body describes in that motion, acted on by no forces either in a vacuum or in a medium with some kind of resistance. Solution. [p. 465] Because the body put in place is not acted on by any absolute forces, the line described by that on the surface is the shortest line in vacuo (62). But the force of resistance in a medium only diminishes the speed of the body and does not affect the direction in any manner; whereby also in a medium with resistance the path described by the body on some surface is equally the shortest. Therefore with the variables in place as before : AP = x, PQ = y and QM = z, (Fig. 91) let dz = Pdx + Qdy be the equation expressing the nature of the surface and Mm , mμ any two elements of the shortest line. From these found above (69) for the shortest line, this is the equation: Thus there arises : But the equation for the surface differentiated gives : with these connected together there is given :EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 702 Therefore with the element Mm given then the following element mμ on the shortest line is found; for it is given by : and the values of ddy and ddz have been found. Whereby hence the position of any following element is determined and the nature of the shortest line by some projection of these is known. Q.E.I. Corollary 1.
834. If, in the equation for the surface P and Q are given in terms of x and y only, then the equation denotes the projection of the shortest line in the plane APQ. [p. 466] Corollary 2.
835. Therefore for the shortest line Mmμ , with the elements selected equally from the axis, then : and from which equations the point μ is known from the two preceding points M and m. Corollary 3.
836. Because the angle RMN vanishes for the shortest line (71), R falls on N; hence the position of the radius of osculation thus is obtained, in order that AX = x + Pz and XR = −Qz − y . Now the length of the radius of osculation (73) is equal to :EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 703 Corollary 4.
837. Now the plane IMR, in which the shortest elements Mmμ are situated, is thus determined, as it becomes : and Now the tangent of the angle, that the plane IMR makes with the plane APQ, is equal to : The secant of this angle is equal to : or the cosine is equal to : Example 1. [p. 467]
838. Let some cylindrical surface have the axis AP; the nature of this is expressed by the equation dz = Qdy with P vanishing in the general equation dz = Pdx + Qdy . Whereby for the projection of the shortest line of this surface in the plane APQ on account of P = 0 and dP = 0 there is obtained this equation: or if indeed Q is only given in terms of y ; but if Q is given in terms of y and z, the variable can be eliminated with the help of the equation dz = Qdy . As in the circular cylinder, in which z 2 + y 2 = a 2 , then Whereby it follows that :EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 704 ∫ Moreover in general, dy (1 + Q 2 ) expresses the arc of the section normal to the axis AP; whereby with the said arc equal to s then αx = s . From which it is understood, if such a surface is set out on a plane, to be the line of the shortest straight line, as agreed. Example 2.
839. Let the proposed surface be some cone having the vertex at A; the equation for such a surface thus can be adapted, so that z is equal to a function of one dimension of x and y. Whereby in the equation dz = Pdx + Qdy the letters P and Q are functions of zero dimensions of x and y. On this account, as now shown elsewhere, it follows that [see E044] : hence [on differentiation] there becomes : [p. 468] and and finally : With which substituted, there is : Put y = px ; P is equal to a certain function of p only, because P is a function of zero dimensions of x and y. Now there is : andEULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 705 From which equation indeed the projection can hardly be recognized. Moreover how the shortest line in such a surface is to be determined, I have set out in more detail in Comment. III. p. 120, [E09 in this series of translations]. Moreover the same as before is to be noted concerning the shortest line, clearly because that set out from the conical surface becomes a straight line in the plane. Scholium.
840. I will not tarry here with the determination in a similar manner of the shortest lines on other forms of surfaces, since in the place cited I have set out this material more fully. Hence I progress to the investigation of the lines which are described on a surface by a body acted on by some forces. Now before this, it is necessary that we examine more carefully the effect of each force. Definition 4.
841. In the following we call the pressing force that normal force, the direction of which is normal to the surface itself in which the body is moving. Corollary. [p. 469]
842. Therefore this pressing force either increases or decreases the centrifugal force, according as the direction of this force falls either opposite to the direction of the radius of osculation of the shortest line, or in that direction (79). Definition 5.
843. In the following we call the force of deflection that normal force, the direction of which is on surface in the tangent plane, and perpendicular to the path described by the body. Corollary.
844. Hence this force deflects the body from the shortest line that the body describes when acted on by no forces, and either draws the body to this or that side [of this line] as the direction of this force either pulls the body this way or that.EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 706 PROPOSITIO 92. Problem.
845. To determine the effect of the pressing force on the body moving on any surface, that is not acted on by any additional forces. Solution. [p. 470] Because this pressing force is normal to the surface and thus the direction of this is in the direction MN, this affects neither the speed nor the direction, as the whole force is taken up on pressing the surface, and therefore the body progresses on the same line on which it was moving if this force were absent; but this is the shortest line determined in the preceding proposition. Therefore the body is moving on the line Mmμ , and the radius of osculation of this MO lies along the normal to the surface MN. Therefore let the direction of this pressing force be MN, which therefore presses the surface inwards along MN. This pressing force is put equal to M; by that the surface is pressed on by a force along MN equal to M. But if the radius of osculation MO is put to lie along the same [undirected] line, then the centrifugal force is contrary to the pressing force, and the effect of this is lessened. Since moreover Mmμ is the shortest line, the radius of osculation is (73) : if twice the height v corresponding to the speed at M is divided by which, then the centrifugal force is produced. On this account the force by which the surface is pressed along MN, is equal toEULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 707 [Note that the radius of curvature has been given as negative above for a concave curve and the sign of this has been reversed, to give a greater force pressing into the curve.] Finally the position of this pressing force has been found previously (68) : AH = x + Pz and HN = −Qz − y , obviously on being sent from the point N, in which the normal MN intersects the plane APQ, with the perpendicular NH to the axis .Q.E.I. Corollary 1.
846. Since no another force is deflecting the normal force, neither a tangential force nor a resistive force if that is present affect the force pressing on the surface, [p. 471], and it is evident from any forces besides acting on the body that the pressing force is always to be of such a size as we have assigned here. Corollary 2.
847. Therefore however great the departure between the path described by the body from the shortest line, the pressing force on the surface is still along the normal to the surface or along the radius of osculation of the shortest line, not along the radius of osculation of the curve described by the body, and neither is the length of this required for the pressing force. Scholium.
848. For that reason we have used that formula of the radius of curvature of the shortest line, in which differentials of the second order are not present, lest these depend on the positions of the two elements Mm and mμ , through which the body is itself moving. But now the radius of osculation must be known from a single element Mm ; for if the body does not describe the shortest line on account of a deflecting force, then differentials of the second order ddy and ddz must be advance, not present in the radius of osculation of the shortest line.EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 708 PROPOSITION 93. Problem.
849. To determine the effect on the motion of the body of the tangential force that pulls the body along the tangent line MT (Fig.92) on some surface. Solution. [p. 472] Let this tangential force be equal to T and the body is progressing through the element Mm with a speed corresponding to the height v ; since this force diminishes the motion, then with the quantities maintaining the same denominations that we have used previously. Now besides this force does not affect either the pressing force nor by the deviation from the shortest line. Now according to the position of the direction of this force, the tangent MT is produced that then crosses the plane APQ at T, then T is a point on the element qQ produced. Therefore and hence From T the perpendicular TF is sent to the axis; then whereby there is obtained : Again since dx : dy = zdx : y − FT then FT = y − zdx , from which the point T is dz dz determined. Q.E.I. Corollary.
850. Since resistance is to be referred to the tangential force, from these it is understood, how the effect is to be determined. For if the resistance is equal to R, thenEULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 709 PROPOSITION 94. Problem.
851. To determine the effect of the normal deflecting force N on a body moving on any surface. Solution. [p. 473] On placing as before AP = x, PQ = y and QM = z (Fig.93) the nature of the surface is expressed by this equation : dz = Pdx + Qdy and the body is moving with a speed corresponding to the height v through the element Mm; in traversing which, unless the deflecting force is present, it proceeds along the element mμ following the shortest line and it gives : and (835). Now the force of the normal deflection N is added, which has the direction against increase. Therefore this force has the effect, that the body in describing the element Mm does not advance to mμ , but is deflected forwards from this direction. Therefore we place it to act along mν ; Mm and mν are two elements of the curve described by the body. Whereby with the perpendicular να sent from ν to the plane APQ then : There is hence obtained : andEULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 710 Now on placing for brevity: the radius of osculation corresponding to the angle between the elements μmν (72) is equal to : Therefore if we call the radius here equal to r, then N = 2rv or 2v = Nr , since here the angle is generated in the same way in which a body in a plane is deflected from a straight line by a normal force. Now it follows that : [p. 474] And in place of ddη and ddζ with the due values substituted, the radius becomes : But since through differentiation of the equation dz = Pdx + Qdy , there is dPdx + dQdy = ddz − Qddy , there is made on substituting this into the equation dz = Pdx + Qdy , on calling Then on this account, Q.E.I.EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 711 Scholium 1.
852. This formula agrees with that which we have found previously (79) in determining the effect of this kind of force. For the difference is only in the sign of the letter N, as it was apparent that the force everywhere had to be taken as negative. And even here we cannot be certain of the sign, because we have extracted it from the root of a squared quantity, and it can be equally positive or negative. Now this doubt, if the calculus is adapted to this special case, is at once removed, because the formula must be of this kind, so that the point ν falls on this side of μ , if the force N is deflecting the body forwards, in order that the direction should be as we have put in place. From which with the help of an example the sign of the square root can also be determined and hence the formula itself found. Corollary 1. [p. 475]
853. If the deflecting force N vanishes, the body continues moving along its own shortest line ; which is indicated by the equation also. For on putting N = 0 there is obtained which is the equation for the shortest line : Corollary 2.
854. Therefore whatever the pressing force and the tangential force and the force of the resistance acting on the body moving on the surface, only if nothing aids the deflecting force, then the body always moves along the shortest line [now called geodesic curves]. Scholium 2.
855. Moreover as concerning the position of this deflecting force N, that can be deduced as follows, since that force is placed on the surface in the tangent plane and likewise it is normal to the curve described, therefore let MG (Fig. 94) be the direction of this force and G is the point at which it crosses the plane APQ , thus so that the force N can be thought to pull along the line MG, while that force we have put before to be deflecting and pressing forwards. Therefore in the first place it is necessary to determine the intersection of the plane of the tangent of the surface at M with the plane APQ, which is the right line TVG ; now this can be found, if two tangents of the surface can be produced as far as the plane APQ, and the points in which they are incident on the plane APQ are joined by a right line. Therefore let MT be the tangent of the described line, which in addition is a tangent of the surface also ; then as we have now established, [p. 476]EULER’S MECHANICA VOL. 2. Chapter 4a. Translated and annotated by Ian Bruce. page 712 (849). Again the surface is understood to be cut by the plane PQM and let MV be the tangent of this cut ; then we have QV = Qz from the equation dz = Pdx + Qdy on putting dx = 0 [or, from the subtangent]. Therefore the point V is known, on account of which the line TV produced is the intersection of the tangent plane of the surface at M with the plane APQ. Therefore the point G, in which the line MG crosses the plane APQ, is placed on the line TV. Again there is taken on the line TQ : and MS is the normal described to the element Mm. And if the normal SG is drawn to QS, from this line SG all the right lines drawn to M are perpendicular to the element Mm. [Thus, the element Mm is normal to the plane MSG.] Whereby since MG is also normal to the element described, the point G is also placed on the line SG. Hence the point G is at the intersection of the lines TV and SG. Now it is the case that : [See annoted Fig. 94] and ang. ELG = ang. PQT. Putting GE = t; then tdy LE = dx and PE = ydy + tdy + zdz . dx Finally also on account of the similar triangles, we have FP : FT + PV = PE : GE − PV , that is : Hence there comes about : and thus the point G is determined. Therefore, if the line QG is drawn, then and

.. and

..