A Treatise on the Theory of Screws

8] TWISTS AND WRENCHES. 11 It a rigid body be acted upon by several wrenches, then these wrenches could be replaced by one wrench which is called the resultant wrench. A twist about a screw a requires six algebraic quantities for its complete specification, and of these, five are required to specify the screw a. The sixth quantity, which is called the AMPLITUDE of the twist, and is denoted by 61, expresses the angle of that rotation which, when united with a translation, constitutes the entire twist. The distance of the translation is the product of the amplitude of the twist and the pitch of the screw, or in symbols, a'pa. The sign of the pitch expresses the sense of the translation corresponding to a given rotation. If the pitch be zero, the twist reduces to a pure rotation around a. If the pitch be infinite, then a finite twist is not possible except the amplitude be zero, in which case the twist reduces to a pure translation parallel to a. A wrench on a screw a requires six algebraic quantities for its complete specification, and of these, five are required to specify the screw a. The sixth quantity, which is called the intensity of the wrench, and is denoted by a > expresses the magnitude of that force which, when united with a couple, constitutes the entire wrench. Ihe moment of the couple is the product of the intensity of the wrench and the pitch of the screw, or in symbols, a" pa. The sign of the pitch expresses the direction of the moment corresponding to a given force. If the pitch be zero, the wrench reduces to a pure force along a. If the pitch be infinite, then a finite wrench is not possible except the intensity be zero, in which case the wrench reduces to a couple in a plane perpendicular n the case of a twisting motion about a screw a the rate at which the amp i tude of the twist changes is called the twist velocity and is denoted by a. 8. Restrictions. It is first necessary to point out the restrictions which we shall impose upon the forces. The rigid body M, whose motion we are considering, is piesuined to be acted upon by the same forces whenever it occupies the same position. The forces which we shall assume are to be such as form what is known as a conservative system. Forces such as those due to a resisting medium are excluded, because such forces do not depend merely on the position of the body, but on the manner in which the body is moving through that position. The same consideration excludes friction which depends on the direction in which the body is moving through the position under considera- tion.