June 26, 1884] 



NA TURE 



207 



In Fig. 1 it is found that there is an uncoiling on the applica- 

 tion of an axial pull. Fig. 2 shows a spring made of the same 

 material, but the wire has been passed through rolls so as to 

 flatten it in the opposite way, and now a rotation tending to coil 

 it up is found to be produced by the application of an axial 

 pull. 



The twisting torque to which the spring is subjected is Frcos a, 

 and the bending torque to Fr sin a. But the twist must be 

 multiplied by sin a, and the bend by cos 

 a when we project these motions on a hori- 

 zontal plane. So far then as the total 

 rotation in a horizontal plane of the free 

 end of the spring relatively to the fixed 

 end is concerned, it may be regarded as 

 being produced by equal twisting and bend- 

 ing torques, each of them equal to F r sin 

 a cos a ; and the total rotation of the free 

 end of the spring relatively to the fixed 

 end, which is the special feature of the 

 springs considered, is proportional to the 

 difference between the two angular rota- 

 tions produced in the wire by these equal 

 bending and twisting torques. The twist 

 alone would cause an increase in the num- 

 ber of coils, that is, a rotation in the direc- 

 tion of coiling which is the positive 

 direction, while the bending, or rather the 

 unbending, alone would cause a negative 

 rotation, or one tending to uncoil the 



spring. When both occur together in the 



actual spiral spring subjected to an axial 



force, the total rotation is positive or nega- 

 tive, according as the angular twist or the 



angular bend is the greater. Hence the 



rlexura and torsional rigidities of the wire 



alone determine whether the rotation is 



positive or negative. 



It is well known, for example, that, 



when a wire of circular section is subjected 



to equal twisting and bending torques, the 



twist is greater than the bending for almost 



all substances, that is, substances in which 



the ratio of the modulus of rigidity to 



Young's modulus is between one-third and 



one-half. Hence we may expect that in 



a spring made of round wire, and with 



the spires making an angle of 45° with a 



plane perpendicular to the axis, the total 



rotation will be positive for an axial force 



applied so as to lengthen the spring. 



And experiment shows that this is the 



case. 



If the wire be flattened and bent so 



that the flat side of the strip touches the 



cylinder on which the wire is coiled, as 



shown in Fig. 1, then the arrangement 



is such that the bending is greater than 



the twist. Hence an axial force applied 



so as to lengthen this spring causes a 



negative rotation, whereas if the strip be 



coiled as in Fig. 2, so that the edge of 



the strip lies against the cylinder on which 



it is coiled, an axial force similarly applied 



large amount of turning of the free end of the spring, combined 

 with small maximum total stress in the material, and not too 

 much axial motion of the free end of the spring, the strip of 

 elliptic section should be as long and as thin as possible, should 

 be wound in a spiral such that the osculating plane makes an 

 angle of 40° to 45° with a plane perpendicular to the axis of the 

 spiral, and so that the smaller diameter of the elliptic section is 

 at right angles to the axis of the spiral. 



wll 



cause a positive rotation. It is 



almost certain that for any strip of material 



the positive value of <p obtained with the 



latter form of spring is likely to be greater 



than the corresponding negative value 



with the former kind, but the difficulty 



of manufacturing the second form of 



spring compelled the authors to confine Fit;. 5. Fig. 6. 



their attention to the former type. 



Having constructed some very delicate springs ,1 the first kind, , In the springs employed by the authors in measuring-instru- 

 ° ne {?* ™ hrs ' amenities which the authors met with arose from ments the edges of the strip nearly touch one another in con- 

 lie liability ot such springs to acquire a permanent set, so that it secutive coils, so that the strip forms almost a continuous 

 1 lecame necessary -to determine the dimensions of the spring which ! cylindric surface, the angle of the spiral being 45°, the cross- 

 would give the largest amount of rotation with the minimum I section of the strip being rectangular, and they find the follow- 

 amount ot stress in the material. Having made their calcula- , ing laws : — 



tion of the greatest amount of stress, the general conclusions ' / F / 1 1 \ 



arm -d at are, that in order, with - given axial force, to obtain a * x /,. ( f N ~ \; ) 



