US3751037A

US3751037A - Arrow fluid-dynamics

Info

Publication number: US3751037A
Application number: US00195554A
Authority: US
Inventors: C Courneya
Original assignee: Individual
Current assignee: Individual
Priority date: 1971-11-04
Filing date: 1971-11-04
Publication date: 1973-08-07
Anticipated expiration: 1990-08-07

Abstract

Disclosed is a new means to cause rotation of a missile as it moves through a fluid, which means maintains the missile in alignment with its direction of movement through the fluid. The means consists essentially of fluid-dynamically spaced elongated depressions in the gross outer surface of at least some portion of the trailing half of the mass of the body part of the missile. These depressions are aligned on the gross outer surface within an angle not greater than 30* from the axis of missile rotation. Each depression has a maximum width and maximum depth less than the length thereof, and also less than the radius of the circumference generated by rotation of the body portion of the missile at the location of the depression. Further, these depressions have a greatest depth portion radially-displaced with respect to a lesser depth portion and this greatest depth portion is forward of the longitudinal midpoint of the depression. The depressions are especially effective to guide the flight of missiles such as arrows, and can be used to replace the normal feathers or fins on standard archery arrows. Each special depression, during transit of the missile through a fluid, creates a resultant fluid-dynamic composite force vector which lies in a line not intersecting the longitudinal axis of rotation of the missile, but passing laterally to that axis. The composite force vector line of each depression passes that axis on the same lateral side, thereby effecting rotation of the missile.

Description

United States Patent 191 Courneya 1 1 Aug. 7, 1973 54 I ARROW FLUID-DYNAMICS [76] Inventor: Calice G. Courneya, Rt. No. 3,

Alexandria, Minn. 56308 [22] Filed: Nov. 4, 1971 [21] Appl. No.: 195,554

[52] US. Cl. 273/l06.5 C, 244/323 [51] Int. Cl. F4lb 5/02 [58] Field of Search 273/1065 C, 106.5 R; 244/323 [56] References Cited UNITED STATES PATENTS 2,828,965 1/1958 Schwitzki 273/l06.5 R

Primary Examiner-Anton O. Oechsle Assistant Examiner-Paul E. Shapiro Att0rneyRobert C. Baker [57} ABSTRACT Disclosed is a new means to cause rotation of a missile as it moves through a fluid, which means maintains the missile in alignment with its direction of movement through the fluid. The means consists essentially of fluid-dynamically spaced elongated depressions in the gross outer surface of at least some portion of the trailing half of the mass of the body part of the missile. These depressions are aligned on the gross outer surface within an angle not greater than 30 from the axis of missile rotation. Each depression has a maximum width and maximum depth less than the length thereof, and also less than the radius of the circumference generated by rotation of the body portion of the missile at the location of the depression. Further, these depressions have a greatest depth portion radially-displaced with respect to a lesser depth portion and this greatest depth portion is forward of the longitudinal midpoint of the depression. The depressions are especially effective to guide the flight of missiles such as arrows, and can be used to replace the normal feathers or fins on standard archery arrows. Each special depression, during transit of the missile through a fluid, creates a resultant fluid-dynamic composite force vector which lies in a line not intersecting the longitudinal axis of rotation of the missile, but passing laterally to that axis. The composite force vector line of each depression passes that axis on the same lateral side, thereby effecting rotation of the missile.

7 Claims, 0 Drawing Figures PATENTEU AUG 7 3,7 5 1 O37 I'NVENTOR. CAL/CE G. COURNEYA A OH/VEY AIRRDW FLUlllD-DYNAMliiCS This invention relates to fluid-dynamics, and more particularly to new missiles characterized by the fluiddynamic features taught herein.

As applied to missles designed for transit through the air, the teaching of this invention may be characterized as that of employing negative air foils, which are distinct from conventional outwardly projecting air foils. Negative air foils are special depressions in the gross surface of a missile, as hereinafter explained in detail.

The teaching of the invention is especially useful in the manufacture of missiles such as arrows for archery use. But the teaching has wide application for missiles designed to be aimed and directed toward a target in a fluid-dynamic manner as they pass through a fluid, whether gas or liquid, air or water. The teaching may be used to make improved torpedos, rifle projectiles, rockets, and the like as well as to make improved arrows and darts. The utility of the invention extends to missiles propelled by a force independent or separate from the missile as well as to those propelled by thrust forces generated in or by the missile.

A primary advantage of the invention is that, for the first time insofar as is known, missiles such as arrows for archery use can be directionally controlled in flight without the problems associated with the use of feathers or this projecting outwardly from the body or shaft of the arrow. Feathers or fins have heretofore been employed as air foils to stabilize the aim and directional flight of arrows. But outwardly projecting elements such as feathers actually also serve to create problems which they are designed to correct. For example, as the outwardly projecting feathers of an arrow pass by the bow in an archery shot, they strike the bow. This serves to kick the trailing end of the arrow slightly out of alignment and away from the bow in a direction not parallel to the intended line of travel for the arrow. immediate correction is necessary to bring the trailing end in line with the direction of travel for the arrow; and the feathers at the end of the arrow are effective to make this correction, but with a loss of forward energy for the arrow. The energy required to correct the instability of flight in the initial few feet of travel serves to slow down the arrow. That energy is lost for the remainder of the distance traveled by the arrow.

In this invention, the missile such as an arrow need not be equipped with outwardly projecting feathers or fins for aerodynamic or fluid-dynamic purposes. Thus, the aforenoted problem created and also corrected (but with loss of the energy of forward motion) by feathers or tins can be obviated.

Depressions in the body of a missile, as taught herein for fluid-dynamic control, are preferably employed without outwardly projecting fluid-dynamic elements such as feathers or fins, but may be, if desired, employed in combination with such outwardly projecting elements. Missiles free of outwardly projecting ele ments and equipped with the fluid-dynamic depressions of this invention are more economical to manufacture. They offer less resistance to movement and travel through a fluid, and thus are capable of attaining or maintaining a relatively greater speed at target impart as compared to otherwise identical but fin equipped missiles. The accuracy of intended direction of flight or movement is relatively greater for missiles using the principles herein as compared to missiles otherwise identical but equipped with fins.

The new missiles of the invention comprise an elongated body portion which is fluid-dynamically rotatable about a longitudinal axis thereof. The mass of the body portion is substantially fluid-dynamically symmetrical in radial planes perpendicular to the axis of rotation; and the gross outer surface of the body portion is fluiddynamically substantially smooth. A leader and a trailer are loacted at opposite ends of the body portion, with at least the leader being fluid-dynamically pointed.

And most important, these new missiles include means effective during transit of the missiles through a fluid to cause rotation of the body portion about the axis of rotation thereof and thereby maintain the missiles in alignment with their direction of movement through the fluid. This means consists essentially of fluid-dynamically spaced elongated depressions in the gross outer surface of at least some portion of the trailing half of the mass of the body portion. The elongated depressions are each aligned on the gross outer surface of the body portion so that each dlepression extends in substantially the same direction with respect to the axis of rotation of the body portion. This directional alignment for the elongated depressions is within an angle not greater than 30 (preferably not greater than 20) from the axis of rotation of the body portion. In the most preferred embodiment of the invention, the directional alignment for the elongated depressions is substantially parallel to the axis of rotation for the body of the missile. Further, the maximum width and maximum depth, of each depression is less than the length thereof, and also less than the radius of the circumference generated or circumscribed by rotating mass of the body portion at the location of the depression. Another characteristic of the depressions is that each has a greatest depth portion which is radially or laterally displaced with respect to a lesser depth portion. in the most preferred embodiment, at least part of the effective lesser depth portion lies within transverse planes (i.e., planes perpendicular to the axis of missile rotation) passing through the portion of greatest depth. However, the greatest depth portion of the depression may be well ahead of (that is, toward the leader end of the missile) the bulk of the lesser depth portion. in all embodiments, the portion of greatest depth of each depression is forward of the longitudinal midpoint of the depression and toward the leader end. During transit of the missile through a fluid, each depression creates a resultant fluid-dynamic composite force vector at its location; and each such composite vector lies in a line not intersecting the longitudinal axis for the rotation of the missile. That line, however, is displaced to pass laterally to the axis of rotation for the missile, with the line for each depression passing on the same lateral side of the axis. I r

The invention will further be described by reference to a drawing, made a part hereof, wherein:

FIG. fl is a schematic plan side view ofa missile characterized as an arrow;

FllGS. 2, 3, l, and 5 are cross-sections through FIG. )1 taken on lines 2-2, 3-44, dl and 5 respectively;

lFllG. 6 is a schematic plan view of a preferred fluiddynamic depression according to the invention, with broken lines of contour in it indicating depth gradients;

H63. '7 and d are schematic plan side views of two of a multitude of alternate shapes for missiles incorporating the teachings herein; and

FIG. 9 is a schematic perspective view of an alternate teaching of the invention.

Referring to the drawing, (especially FIGS. 1-5 inclusive), missile 10 comprises an elongated body portion 11 between a leader l2 and a trailer 13. The leader 1'2 and trailer 13 may simply be opposite ends of the body portion of the missile in the sense that they may be formed of the same material and be unitary with the body portion. Thus, transverse dash lines 14 and 15 in FIGS. 1 and 5 are for the purpose of placing an approximate boundary separating the body 11 from the leader or header 12 (that is the leading end of the missile) and from the trailer 13 (that is, the trailing end or stern of the missile). It will be understood that leader 12 and trailer 13 may comprise entirely separate pieces from 'the body 11, and may or may not be joined to each other by means of a shaft or the like separate from the skin or surface member of the body portion. Preferably both the header l2 and trailer 13 rotate with the rotatable body ll, but this is not always required.

The elongated body portion (in particular, the skin or suface structure thereof) is fluid-dynamically rotatable about a longitudinal axis of the body portion. Where the body portion is cylindrical or circular in gross configuration, and its rotating mass is substantially uniform throughout, the axis ofrotation under fluid-dynamic conditions will substantially coincide with the axis of its cylindrical or circularconfiguration. In all cases, however, the body portion is rotatable about a longitudinal axis which may be more or less fixed under fluiddynamic conditions by the nature of its outer skin contour and any variation in'mass or density in different parts of the rotatable'body portion.

Another characteristic of the rotatable body portion is that its mass is substantially fluid-dynamically symmetrical in radial planes perpendicular to its rotation axis. This does not require the body surface to be purely cylindrical or circular in cross section. It may approximate an oval or triangle or square or still other shapein cross-sections taken at different locations along its length. But the rotatable body must be substantially fluid-dynamically symmetrical in the sense that its mass distribution and shape contours permit it to rotate on the fixed axis without generating any substantial amount of wobble as it passes through a fluid. Preferably, it is geometrically symmetrical as well as fluid-dynamically symmetrical in planes through its axis of rotation.

Also, the gross outer surface of the body portion should be fluid-dynamically substantially smooth. The ordinary concept known to designers as streamlining satisfies this criteria.

The leader 12 should be pointed in a fluid-dynamic or streamlined sense; and preferably the streamlined point substantially coincides with the axis of rotation for the body portion. The trailer 13 need not be curved or pointed, but may be fluid-dynamically pointed. As illustrated in FIG. I, the trailer 13 may comprise a nock or grooved part for reception of the bowstring of a bow. Either or both the leader l2 and trailer 13 may optionally be radially larger or smaller than immediately adjacent portions of the body 11.

Description now will be devoted to the depressions broadly designated as 16 in FIGS. l-6. These depressions form the means, effective during transit of the missile through a fluid, to cause rotation of the body portion 11 about an axis as aforenoted. Thus the depressions, by fluid-dynamically causing rotation, serve to maintain the elongated missile in alignment with its direction of movement through the fluid.

Depressions 16 are elongated and are fluiddynamically spaced in the gross outer surface of the body portion. They may optionally be spaced over the entire body portion; but they are at least present in some portion of the trailing half of the mass of the rotatable body portion. Where the mass of the rotatable body portion is substantially uniform and a cylindrical or circular body is employed, the spacing for fluiddynamic purposes is approximately in equal geometric radial increments about the body.

In FIGS. l-5 inclusive this is illustrated by equidistant spacing of three depressions 16. Variation from equidistant spacing is desirable where the mass or contour of the rotatable body 11 varies. The fundamental principle is that the spacing must be such that fluid flow action on the plurality of depressions is balanced consistent with the mass and contour of the rotatable body 11. The concept again is one where the fluid-dynamic forces acting through the depressions. on the body to cause rotation of the body are substantially symmetrically balanced. Preferably, at least three depressions are employed. Thus a plurality of depressions are common. A multiplicity of depressions (over 10 or 20 or so many as to make it impractical toengage in counting) may be desirable in some cases.

The elongated.,depressions' 16 are aligned on the gross outer surfaceof the body 1 so that they each extend in substantially the same direction with respect to the axis of rotation for the body 11. The direction of alignment is within an angle not greater than 30 from the axis, and preferably not greater than 20. As illustrated in FIG. 1, the most preferred alignment is that substantially parallel to the axis of the shaft comprising body 11. Asubstantially'parallel alignment, as illustrated, contributes to minimal generation of turbulence acting against those desired forces of the fluid on the depressions. Of course, some turbulance of fluid flow in the area of the depressions is an inherent feature of the invention; but the desired force components effected by fluid flow over the depressions are proportionally reduced as the alignment orientation of the preferred depressions is changed from that orientation which is substantially parallel to the rotatlonal axls 0F body 11. Thus orientation at an angle from about 5 or l0 up to 20 or 30 from the axis is accompanied by a gradual and, in some cases, substantial decrease of the effective force components desired, with the result that the effective force components for rotation of the body become minimal or quite unreliable at angles over about 30. Also, orientation at significant angles tends to provide results which vary depending upon speed of missile transit, which creates a problem of predictability.

Two important characteristics of each depression are as follows: the maximum width and maximum depth of each depression are less, in each instance,- than the length of the depression. The maximum width and maximum depth also are less than the radius of the circumference generated by or circumscribed by rotation of the body portion at the location of the depression. These features contribute to minimizing the drag effect which inherently accompanies depressions (but is normally less than the drag effect of projecting fins or feathers).

Depressions I6 each are characterized further by having a portion which, under fluid-dynamic conditions of missile movement through a fluid, usually effects a localized reduction in pressure; and this portion may be characterized as the vacuum" portion of the depression. It is normally, but not necessarily always, the portion of maximum or greatest depth for the depression. The greatest depth portion is illustrated in FIG. 6 by the boundary established by the borken line of pressure gradient about the space identified by numeral 117. This portion of greatest depth, is always forward, preferably substantially forward, of the longitudinal midpoint (identified by dash line 118 through FIG. ti) of the depression. It is toward the leading end of the missile from the midpoint 18. Further, portion 17 preferably is lateral to the width midpoint (identified by dash line W through FIG. 6) of the depression.

Each depression also has what might be called a pressure portion, although the term pressure" may be somewhat of a misnomer and is solely a designation or term to indicate higher pressure than that characteristic for the effective vacuum portion. In the preferred alignment of depressions parallel to the axis of rotation of body II, this pressure portion or portion of lesser depth is identified by the shaded portion 2@ in FIG. 6. It is in large part radially lateral and generally rearward of the greatest depth portion I7. It is always radially displaced from the greatest depth portion t7.

At this point, the behavioral relationship between the greatest depth portion and lesser depth portions of depression llti will be considered for the preferred embodiment where the special depressions are aligned substantially parallel to the axis of rotation of body II. Under such conditions, the greatest depth portion ll? behaves as a vacuum portion and lesser depth portion 2t) as a pressure" portion, as the missile moves fluid-dynamically through a fluid. The force effect of this is such that the vector of resultant force is other than in a direct radial line from the axis of the missile body rotation. The resultant composite fluid-dynamic force vector lies in a line which does not intersect the axis of rotation. That line passes laterally to the missile axis of rotation. The composite force vector lines for a missile with depressions oriented as illustrated in FIGS. 1-5 inclusive are schematically shown by outwardly directed straight arrow lines in FIG. 3 of the drawing. The force vector lines pass on the same relative lateral side of the axis of rotation, so that the effective force of each depression augments (instead of opposes) the effective force of the other depression. Thus it is that the body portion of the missile is caused to be torqued or rotated, during transit thorugh a fluid, in a direction causing the pressure portion 20 to trail the vacuum" portion 17 of the depressions (see FIGS. 1-45, inclusive). The tail end 21 of teh depressions is of shallow depth, and is designed to reduce the drag effect of fluid acting in that area as the missile is passed through the fluid.

Substantially similar results are achieved by varying the angle of orientation for the preferred depressions (i.e., those illustrated in FIGS. Il-d, inclusive) a modest amount such as 5 or so from the substantially parallel alignment. However, as the angle is varied (to 5 from parallel), the shape of the effective vacuum area or part 117 of the greatest depth portion will be altered, as will also the shape of the effective pressure area or part 20 of the lesser depth portion. The effect at higher angles from parallel (especially when the angular orientation of the depression is moved in a clockwise direction for the depression In as viewed in FIG. ll) tends to substantially change the functional performance of the areas I7 and 20. One can even cause reversal of the direction of rotation of the missile from that discussed and illustrated in FIGS. 1-5 inclusive. In short, the functional effect of the contour of the greatest depth portion and lesser depth portion, in terms of fluiddynamic phenomena, will vary as orientation is altered from the preferred parallel for the depressions. Further, the showing in the drawing should not be taken as literal for these areas. It is a schematic showing. Nevertheless, in all cases, the fluid-dynamic phenomena effected at the location of the depressions is such as to create a,fluid-dynamic composite force vector. And this composite vector does not coincide with a radial line intersecting the axis of the missile rotation. It is this feature which causes the torque rotational effect as the missile passes through fluid.

The illustrative missile in FIG. '7 comprises a body 21 of varying diameter along its length, with a leader or header end 22 which is fluid-dynamically pointed (but not geometrically pointed), and a trailer 23 which is geometrically pointed. The missile of FIG. h comprises a body portion 2d, a narrow neck part 25, an enlarged header 2b, and a trailer 27 having fins 2% projecting outwardly. These shapes and others are possible for missiles utilizing principles of the invention.

The preferred depressions as aforediscussed are suitably molded in the body portion, that is in the skin member or outer surface of the body portion, during manufacture of the missile.

A less preferred but economical approach to the manufacture of missiles which incorporate many basic concepts of this invention is illustrated in FIG. 9. In that figure, the body 29 of the missile is placed in the space framed by substantially parallel edges of masking plates 3th and 3t. Body 29 is then slowly rotated as granules or small beads (e.g., glass beads of 1-10 mils diameter) are impinged at high velocity on the exposed strip of body 29 between plates 3ft and fill. Impingement is accomplished at a slight angle, up to about 20 or 30 from the axis direction of body 29. The granules or beads employed must possess a hardness value significantly above the material chosen for the body 29 (or skin thereof). Also the force of impingement upon the body must be great enough to cause the impinging ma terial to dent the skin of body 29 and form a tear-drop or substantially similar shape as a depression in the gross outer surface of body 29. The resulting tear-drop depressions are oriented or aligned, as aforediscussed for the preferred depressions, within 30 from the axis of rotation of the missile. Tear-drop depressions are not suitably aligned in a purely parallel. manner to the axis of rotation. They must be at least a couple degrees from parallel up to the angle of about 20 or 30. The vacuum" portion of greatest depth in teardrops is located in the blunt portion of the tear-drop (opposite the tail), whereas the pressure portion of lesser depth is in that portion known as the tail of the tear-drop. Teardrop depressions cause considerably turbulence at the location of the tear-drop, with resultant drag. However, since the depressions are so minute, the total drag effect is less noticeable than one might surmise. Nevertheless, tear-drops are relatively inefficient as compared to the preferred depressions, but even so, they are useful where economy of manufacture is the prime consideration. Interestingly, the rotation effected by tear-drops has been found in some cases to be the reverse of that discussed for the preferred depressions. This may possibly be explained by the fact that the vacuum portion of greatest depth in the minute teardrops is indeed very shallow, with the result that the vacuum or reduced pressure force component generated by it is so weak that the force component generated at the pressure portion of the tear-drop (which is at the tail portion of the tear-drop) seems to serve as the primary factor dominating the composite force vector of the depression.

That which is claimed is:

1. A missile comprising (a) an elongated body portion fluid-dynamically rotatable about a longitudinal axis thereof, the mass of said body portion being substantially fluid-dynamically symmetrical in radial planes perpendicular to said axis, and the gross outer surface of said body portion being fluid-dynamically substantially smooth, (b) a leader and trailer at oppo-' site ends of said body portion, at least said leader being fluid-dynamically pointed, (c) and means effective during transit of said missile through a fluid to cause rotation of said body portion'about said axis and thereby maintain the missile in alignment with its direction of movement through the fluid, said means consisting essentially of fluid-dynamically spaced elongated depressions in the gross outer surface of at least some portion of the trailing half of the mass of said body portion, each'saiddepression (i) being aligned on said gross outer surface'within an angle not greater than 30 from said axis, (ii) having a maximum width and maximum depth less than the length thereof, and less than the radius of the circumference generated by rotation of said body portion at the location thereof, and (iii) having a greatest depth portion radially displaced with respect to a lesser depth portion thereof, said greatest depth portion being forward of the longitudinal midpoint thereof and toward the leader of said missile, whereby each said depression during said transit of said missile through a fluid creates a resultant fluid-dynamic composite force vector which lies in a line not intersecting the aforesaid longitudinal axis of said body portion but passing laterally thereto, each said line passing on the same relative lateral side of said axis.

2. The missile of claim 1 wherein said greatest depth portion of the depressions is lateral to the width midpoint of the depressions.

3. The missile claim 1 wherein said depressions are asymetrical in linear contour in radial planes perpendicular to the axis of said body portion.

4. The missile of claim 1 wherein said depressions are tear-drop shapes in outer perimeter.

5. The missile of claim 1 wherein said depressions are minute in size and a multitude of said depressions are present in the gross surface of said body portion.

6. The missile of claim 1 wherein the directional alignment of said elongated depressions is substantially parallel to said axis of rotation.

7. The missile of claim 1 wherein body portion comprises a straight shaft and the trailing end includes a groove adapted to receive a bowstring.

- a: a a: a:

a UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION latent No. 3,751,037 Dated August 7; 197? Inventor(s) Calice COuTneYa- It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

Colt-man 4, line 46, "rotational axIs 0F I should read 11 rotational axisof Column 5, line 9, "borken" should read Broken vw; line 55, teh should read the -r; lines 63 and 64 "(to 5 from parallel)" should read t 5 and above 5 from parallel) Column 8, line 18, "missile claim l should read missile of claim 1 Signed and sealed this 8th day of January .1974.

Attest EDWARD M. FLETCHER,'JR. RENE D. TEGTMEY E R Attesting Officer Acting Commissioner of Patents 3 row Po-1o so (10-69) uscom -pc sous-pan UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION a Patent No. 3,751 ,037 Dat d August 7, 1973 Invent0r(s) Calice G. Courneya It is certified that e rrof appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

Column 4, line 46, "rotational ends 01 should read rotational axis .of Column 5, line 9, -'boTken" should read broken line 55, "teh" should 'read the lines 63 and 64 (to 5 from parallel)" should read to 5 and above 5 from parallel) Column 8 line 18 "missile claim 1 should read missile of claim 1 Signed and sealed this 8th day of Januaryhl974.

L E T Attest; V

EDWARD M. FLETCHER,JR. t RENE D." TEGTMEYER Attesting Officer u Acting Commissioner of Patents

Claims

1. A missile comprising (a) an elongated body portion fluiddynamically rotatable about a longitudinal axis thereof, the mass of said body portion being substantially fluid-dynamically symmetrical in radial planes perpendicular to said axis, and the gross outer surface of said body portion being fluid-dynamically substantially smooth, (b) a leader and trailer at opposite ends of said body portion, at least said leader being fluiddynamically pointed, (c) and means effective during transit of said missile through a fluid to cause rotation of said body portion about said axis and thereby maintain the missile in alignment with its direction of movement through the fluid, said means consisting essentially of fluid-dynamically spaced elongated depressions in the gross outer surface of at least some portion of the trailing half of the mass of said body portion, each said depression (i) being aligned on said gross outer surface within an angle not greater than 30* from said axis, (ii) having a maximum width and maximum depth less than the length thereof, and less than the radius of the circumference generated by rotation of said body portion at the location thereof, and (iii) having a greatest depth portion radially displaced with respect to a lesser depth portion thereof, said greatest depth portion being forward of the longitudinal midpoint thereof and toward the leader of said missile, whereby each said depression during said transit of said missile through a fluid creates a resultant fluid-dynamic composite force vector which lies in a line not intersecting the aforesaid longitudinal axis of said body portion but passing laterally thereto, each said line passing on the same relative lateral side of said axis.