Articulated mannequin with various assembly mechanisms

Abstract

This invention relates to an articulated mannequin, comprising various assembly mechanisms that keep the limbs thereof assembled, in which said limbs may be rigid or flexible as well as hollow or solid. Said assembly mechanisms include, at least: magnetic connection mechanisms, magnetic locking and connection mechanisms, adjustable locking and connection mechanisms, adjustable twisting and connection mechanisms, free movement and connection mechanisms, automated movement and connection mechanisms, compound disc mechanisms, double ball-and-socket joint mechanisms and compression ball-and-socket joint mechanisms. Said mechanisms are characterized in that they are inserted in the material of the limbs of the articulated mannequin, and in that they give the articulated mannequin a complete range of movement and versatility, enabling it to take up any human posture and be installed in almost any location. By virtue of said assembly mechanisms, the limbs of the articulated mannequin can: remain static in a given position; return automatically to a given position, after having been moved; be assembled around a dividing sheet; perform automated movements; and perform movements without resistance, when interacting with moving objects; thereby being able to be put to more uses, in the location where the articulated mannequin is positioned.

1 - 74 . (canceled) 
     
     
         75 . An articulated mannequin, comprising:
 i. a head ( 2 ), comprising a vertical cavity ( 2   a ) located on its lower portion, which in turn comprises an orifice ( 2   b ) located in its upper portion;   ii. a torso ( 3 ), comprising a neck ( 3   c ) vertically inserted into the cavity ( 2   a ) of the head ( 2 ), wherein said neck ( 3   c ) besides being cylindrical at least in its upper portion, also comprises a vertical cavity ( 3   d ) concentrically located in its interior, a concentrically perforated ironwork ( 93 ) which is wider at its inferior portion, is inserted under the cavity ( 3   d ) of the neck ( 3   c ), additionally that a screw ( 92 ) is fixed in the central orifice of said ironwork ( 93 ); wherein the torso ( 3 ) further comprises two horizontal cavities ( 3   e ) located at the lateral sides of its upper portion, whose lower parts end on a flat surface ( 3   f );   iii. at least one double kneecap mechanism ( 91 ′) joining the head ( 2 ) with the torso ( 3 ), which is located inside the cavities ( 2   a  and  3   d ) of the head ( 2 ) and of the neck ( 3   c ), where each kneecap ( 91 ) of said mechanism ( 91 ′) comprises at least: a spherical protrusion ( 91   b ), a spherical cavity ( 9   c ) and a central perforation ( 91   e ) passing longitudinally therethrough, where a first kneecap ( 91 ) is fixed to the screw ( 92 ) inserted in the cavity ( 3   d ) of the neck ( 3   c ), wherein said screw ( 92 ) is inserted through the central perforation ( 91   e ) of said kneecap ( 91 ), while, a second kneecap ( 91 ) is joined with the first kneecap ( 91 ), thus wrapping by means of its spherical cavity ( 91   c ) the spherical protuberance ( 91   b ) of said first kneecap ( 91 ), and one end of the ends of a rod ( 94 ) is inserted into the center perforation ( 91   e ) of the second kneecap ( 91 ), in addition that the other end of said rod ( 94 ) is inserted into the orifice ( 2   b ) of the cavity ( 2   a ) of the head ( 2 );   iv. at least one superior hemispherical connecting piece ( 1 ), horizontally inserted into either horizontal cavities ( 3   e ) of the torso ( 3 ), which is formed by at least a closed half sphere portion ( 62 ) and a preferably semi-cylindrical or semi-conical closed portion ( 63 ), both joined by a transition portion ( 64 ), so that the flat surfaces ( 65  and  66 ) of both portions ( 62  and  63 ) are arranged at least perpendicularly to each other, wherein the flat surface ( 65 ) of the semi-cylindrical or semi-conical closed portion ( 63 ), is making contact with the flat surface ( 3   f ) of the horizontal cavities ( 3   e ) of the torso ( 3 );   v. a first magnetic joining mechanism ( 37 ) that connects the torso ( 3 ) with the upper hemispherical connecting piece ( 1 ), wherein said magnetic joining mechanism ( 37 ) comprises at least two smooth disks ( 20 ) which are parallel to each other, one of which is inserted on one flat surface ( 3   f ) of the horizontal cavity ( 3   e ) of the torso ( 3 ) and the other is inserted on the flat surface ( 65 ) of the semi-cylindrical or semi-conical closed portion ( 63 ) of the upper hemispherical connecting piece ( 1 ), wherein said smooth disks ( 20 ) comprise attaching means which are immersed in the material of the members to be joined, thus leaving a flat surface ( 20   a ) exposed and aligned with the flat surface ( 65  or  66 ) of the member in which they are located, and at least one magnet ( 38 ) which is concentrically attached to one of the smooth disks ( 20 ) on its flat surface ( 20   a ), where the anchoring means are projected;   vi. at least one arm ( 5 ) comprising at one of its ends, a half sphere closed portion ( 62 ) and a portion that is similar to the anatomical shape of an arm, both joined by a transition portion ( 64 ), where the flat surface ( 66 ) of its closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the upper hemispherical connecting piece ( 1 ); and   vii. a second magnetic joining mechanism ( 37 ) that connects the arm ( 5 ) with the upper hemispherical connecting piece ( 1 ), wherein one of the smooth disks ( 20 ) is inserted into the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the upper hemispherical connecting piece ( 1 ), and the other flat disk ( 20 ) on the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the arm ( 5 ).   
     
     
         76 . The articulated mannequin according to  claim 75 , wherein the torso ( 3 ) further comprises an upper torso ( 3   a ) and a lower torso ( 3   b ), the mannequin further comprising a magnetic joining mechanism ( 37 ) at a joining point ( 10   a ) between the upper torso and the lower torso. 
     
     
         77 . The articulated mannequin according to  claim 75 , wherein the arm ( 5 ) further comprises a biceps ( 5   a ), an elbow ( 5   b ), a forearm ( 6   a ), and a hand ( 6   b ), the mannequin further comprising: a magnetic joining mechanism ( 37 ) at joining points ( 12 ,  13 ,  81 ) between the biceps, the elbow the forearm and the hand. 
     
     
         78 . The articulated mannequin in accordance with  claim 75 , wherein the arm ( 5 ) comprises at least: a biceps ( 5   a ), elbow ( 5   b ), forearm ( 6   a ), hand ( 6   b ) and all possible combinations between them, for which, it is added a magnetic joining mechanism ( 37 ) at joining points ( 12 ,  13 ,  81 ) between them; and
 the biceps ( 5   a ), the elbow ( 5   b ) and the forearm ( 6   a ) have a configuration similar to the one of the upper hemispherical connecting piece ( 1 ), with the difference that its semi-cylindrical or semi-conical closed portion ( 63 ) resembles, at least, the anatomical shape of a biceps ( 67 ), an elbow and a forearm respectively.   
     
     
         79 . The articulated mannequin according to  claim 75 , additionally comprising a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ). 
     
     
         80 . The articulated mannequin according to  claim 75 , additionally comprising, a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ); wherein the lower portion ( 9   b ) comprises, at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   iv. at least one magnetic joining mechanism ( 37 ), as described in  claim 75 .   
     
     
         81 . The articulated mannequin according to  claim 75 , additionally comprising, a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ); wherein the lower portion ( 9   b ) comprises, at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   iv. at least one magnetic joining mechanism ( 37 ), as described in  claim 75 , wherein the magnetic joining mechanism ( 37 ) are located in joining points ( 15 ) between the horizontal cavities ( 4   a ) of the hip ( 4 ) and the lower hemispherical connecting pieces ( 1 ), and at the joining points ( 16 ) between said lower hemispherical connecting pieces ( 1 ) and the legs ( 7 ).   
     
     
         82 . The articulated mannequin according to  claim 75 , additionally comprising, a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ); wherein the lower portion ( 9   b ) comprises, at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   iv. at least one magnetic joining mechanism ( 37 ), as described in  claim 75 ;   
       wherein the legs ( 7 ) comprising at least: an upper thigh ( 7   a ), a lower thigh ( 7   b ), a calf ( 8   a ), a foot ( 8   b ) and all possible combinations between them, for which, it is added a mechanism magnetic joining mechanism ( 37 ) at joining points ( 17 ,  18 ,  82 ) between them. 
     
     
         83 . The articulated mannequin according to  claim 75 , additionally comprising, a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ); wherein the lower portion ( 9   b ) comprises, at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   iv. at least one magnetic joining mechanism ( 37 ), as described in  claim 75 ;   
       wherein the legs ( 7 ) comprising at least: an upper thigh ( 7   a ), lower thigh ( 7   b ), a calf ( 8   a ), a foot ( 8   b ) and all possible combinations between them, for which it is added a magnetic joining mechanisms ( 37 ) at joining points ( 17 ,  18 ,  82 ) between them and wherein the upper thigh ( 7   a ), the lower thigh ( 7   b ) and the calf ( 8   a ) have a similar configuration to the lower hemispherical connecting pieces ( 1 ), with the difference that its semi-cylindrical or semi-conical closed portion ( 63 ) resembles at least the anatomical shape of the upper portion of a leg, of a thigh and of a calf respectively. 
     
     
         84 . The articulated mannequin according to  claim 75 , wherein the arm ( 5 ) comprises at least: a biceps ( 5   a ), en elbow ( 5   b ), a forearm ( 6   a ), a hand ( 6   b ) and all possible combinations among them, for which it is added a joining mechanism at the joining points ( 12 ,  13  and  81 ), which includes at least one joining mechanism selected from the group comprising, but not limited to: a joining and simple free movement mechanisms; a joining and free movement mechanisms for working with radial loads, an adjustable torsion and joining mechanism, a joining and free movement mechanisms for working with axial loads; a magnetic locking and joining mechanism; and adjustable locking and joining mechanisms; an adjustable double locking and joining mechanism; an adjustable locking and joining mechanism having a movable screw. 
     
     
         85 . The articulated mannequin according to  claim 75 , additionally comprising, a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ); wherein the lower portion ( 9   b ) comprises, at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   iv. at least one magnetic joining mechanism ( 37 ), as described in  claim 75 ;   
       wherein the legs ( 7 ) comprising at least: an upper thigh ( 7   a ), lower thigh ( 7   b ), a calf ( 8   a ), a foot ( 8   b ) and all possible combinations between them, for which it is added a magnetic joining mechanisms ( 37 ) at joining points ( 17 ,  18 ,  82 ) between and wherein it is added a joining mechanisms for the joining points ( 15 ) between the horizontal cavities ( 4   a ) of the hip ( 4 ) and the lower hemispherical connecting pieces ( 1 ), and at the joining points ( 16 ) between said lower hemispherical connecting pieces ( 1 ) and legs ( 7 ), and wherein it is included at least one attachment mechanism selected from the group comprising, but not limited to: a joining and simple free movement mechanisms; a joining and free movement mechanisms for working with radial loads, an adjustable torsion and joining mechanism, a joining and free movement mechanisms for working with axial loads; a magnetic locking and joining mechanism; and adjustable locking and joining mechanisms; an adjustable double locking and joining mechanism; an adjustable locking and joining mechanism having a movable screw. 
     
     
         86 . The articulated mannequin according to  claim 75 , wherein at least one of the disks ( 20 ) also called simple disks of the assembly mechanisms, is replaced by a composite disc, which is comprised by at least:
 i. A coupling disc ( 31 ′), which is similar to said simple disks, with the difference that it lacks a flange ( 20   b ), additionally comprising on its flat surface ( 20   a ), at least one conical hole ( 32 ) equidistant between them, located around its central orifice ( 20   c ,  20   d  and  36   a );   ii. an insertion base ( 30 ) comprised by at least: a flat surface ( 30   a ), which comprises attaching means which are immersed in the material of the members to be joined, leaving said flat surface ( 30   a ) exposed and aligned with the flat surface ( 65  or  66 ) of the member in which it is located, wherein the flat surface ( 30   a ) of the insertion base ( 30 ) comprises at least one central orifice ( 30   c ) of equal or larger diameter than the central orifice ( 20   c ,  20   d  or  36   a ) of the coupling disc ( 31 ′), and more than one threaded orifices ( 30   e ) aligned with the conical holes ( 32 ) of the coupling disc ( 31 ′), wherein said coupling disk ( 31 ′) is concentrically positioned on the flat surface ( 30   a ) of the insertion base ( 30 ), with their conical holes ( 32 ) aligned with the threaded holes ( 30   e ) of the insertion base ( 30 );   iii. at least one screws having a conical head ( 56 ), which are inserted through conical holes ( 32 ) of the coupling disc ( 31 ′), until they are screwed into the threaded holes ( 30   e ) of the insertion base ( 30 ), wherein the head of said screws having a conical head ( 56 ), is housed within the cavity of said conical orifices ( 32 ).   
     
     
         87 . The articulated mannequin according to  claim 75 , additionally comprising, a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ); wherein the lower portion ( 9   b ) comprises, at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   iv. at least one magnetic joining mechanism ( 37 ), as described in  claim 75 ;   
       wherein the simple disks and/or the coupling disks ( 31 ′) comprise at least one device such as a mechanical, electrical, electronic, pneumatic and/or hydraulic device which is bonded to its flat surface ( 20   a ) of the side that remains inside the member in which it is located. 
     
     
         88 . The articulated mannequin according to  claim 75 , additionally comprising, a lower section ( 9   b ) that is comprised by a single member, which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), by at least one magnetic joining mechanism ( 37 ); wherein the lower portion ( 9   b ) comprises, at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   iv. at least one joining mechanism;   
       wherein the joining mechanism comprises a joining and automated movement mechanism ( 55 ), which is comprised by at least: a composite disk, where a motor ( 35 ) is concentrically placed on the flat surface ( 20   a ) of its coupling disk ( 34 ), additionally, the axle ( 35   a ) of the motor ( 35 ) comprises a thread, at least in a part of its cylindrical section, wherein said axle ( 35   a ) passes through the central orifice ( 20   a ) of the coupling disk ( 34 ) of said composite mechanism; a disk having a central guide orifice or a second composite disk, wherein said axle ( 35   a ) also passes through the central orifice ( 36   a ) of the disk having a central guide orifice or through the coupling disc ( 36 ) of the second composite disk, until it is screwed into a nut having nylon insert ( 57 ), located on the side where the flanges are projected ( 20   b  or  30   b ) of the disk having a central guide orifice or of the insertion base ( 30 ) of the second composite disk, as appropriate. 
     
     
         89 . The articulated mannequin according to  claim 75 , wherein the arm ( 5 ) comprises at least: a biceps ( 5   a ), elbow ( 5   b ), forearm ( 6   a ), hand ( 6   b ) and all possible combinations among them; it is provided a lower section ( 9   b ), which extends from the hip ( 4 ) until the foot ( 8   b ), wherein said lower portion ( 9   b ) remains joined to the lower torso ( 3 ), and is comprised by at least:
 i. a hip ( 4 ), which in turn comprises two horizontal cavities ( 4   a ) located at the lateral sides of the lower portion, whose lower portions end in a flat surface ( 4   b );   ii. two lower hemispherical connecting pieces ( 1 ) similarly comprised as the upper hemispherical connecting pieces ( 1 ) described in  claim 75 , wherein each one is horizontally inserted into the horizontal cavities ( 4   a ) of the hip ( 4 );   iii. two legs ( 7 ), similarly comprised as the arm ( 5 ) described in  claim 75 , with the difference that the portion that resembles the anatomical shape of an arm, in this case is similar to the anatomical shape of one leg, wherein the flat surface ( 66 ) of the closed half sphere portion ( 62 ), is making contact with the flat surface ( 66 ) of the closed half sphere portion ( 62 ) of the lower hemispherical connecting pieces ( 1 );   
       wherein the legs ( 7 ) comprising at least: an upper thigh ( 7   a ), a lower thigh ( 7   b ), a calf ( 8   a ), a foot ( 8   b ) and all possible combinations between them; 
       it is included compression kneecap mechanism ( 80 ), which is formed by at least: a member ( 6   b  or  8   b ) with a partially spherical protrusion ( 83 ) with a conical cavity ( 84 ), a washer ( 86 ) with a thread at its central orifice, which concentrically positioned under said conical cavity ( 84 ), being inserted between the material of said member ( 6   b  or  8   b ); an eyebolt ( 87 ) is introduced through the conical cavity ( 84 ), until it is screwed into the threaded washer ( 86 ); a compression spring ( 88 ) with a hook at each of its ends, is inserted through the circular orifice ( 85   a ) of the flat circular surface ( 65 ) of a member ( 6   a  or  8   a ), until one of its hooks is inserted into at least one perforation ( 89 ) of the flange ( 20   b ,  26   f ,  30   b ) of a simple disk, of an ironwork having flange ( 26   e ) or of an insertion base accordingly, additionally the second hook of the compression spring ( 88 ), is engaged in the eyebolt ( 87 ), wherein finally, the partially spherical protuberance ( 83 ) of the first member ( 6   b  or  8   b ), is partially inserted into the circular orifice ( 85   a ) of the flat circular surface ( 65 ) of the second member ( 6   a  or  8   a ).