116 THE THEORY OF SCREWS. [128, 



on which would make the body commence to twist about 6, is indeterminate. 

 Any screw in space which is reciprocal to &amp;lt;/&amp;gt; would fulfil the required condition 

 (136). 



We have seen in 96 that an impulsive wrench on any screw in space may 

 generally be replaced by a precisely equivalent wrench upon the cylindroid 

 which expresses the freedom. We are now going to determine the screw 77, 

 on the cylindroid of freedom, an impulsive wrench on which would make the 

 body twist about a given screw 6 on the same cylindroid. This can be easily 

 determined with the help of the pitch conic ; for we have seen ( 40) that a 

 pair of reciprocal screws on the cylindroid of freedom are parallel to a pair 

 of conjugate diameters of the pitch conic. The construction is therefore as 

 follows: Find the diameter A which is conjugate, with respect to the ellipse 

 of inertia, to the diameter parallel to the given screw 6. Next find the 

 diameter B which is conjugate to the diameter A with respect to the pitcfi 

 conic. The screw on the cylindroid parallel to the line B thus determined 

 is the required screw 77. 



Two concentric ellipses have one pair of common conjugate diameters. 

 In fact, the four points of intersection form a parallelogram, to the sides of 

 which the pair of common conjugate diameters are parallel. We can now 

 interpret physically the common conjugate diameters of the pitch conic, and 

 the ellipse of inertia. The two screws on the cylindroid parallel to these 

 diameters are conjugate screws of inertia, and they are also reciprocal; they 

 are, therefore, the principal screws of inertia, to which we have been already 

 conducted (127). 



If the distribution of the material of the body bear certain relations to 

 the arrangement of the constraints, we can easily conceive that the pitch 

 conic and the ellipse of inertia might be both similar and similarly situated. 

 Under these exceptional circumstances it appears that every screw of the 

 cylindroid would possess the property of a principal screw of inertia. 



129. The Ellipse of the Potential. 



We are now to consider another ellipse, which, though possessing many 

 useful mathematical analogies to the ellipse of inertia, is yet widely different 

 from a physical point of view. We have introduced ( 102) the conception 

 of the linear magnitude w , the square of which is proportional to the work 

 done in effecting a twist of given amplitude about a screw a. from a position 

 of stable equilibrium under the influence of a system of forces. We now 

 propose to consider the distribution of the parameter v a upon the screws of 

 a cylindroid. It appears from 102 that if v l} v denote the values of the 

 quantity v a for each of two conjugate screws of the potential, and if a,, 2 

 denote the intensities of the components on the two conjugate screws of a 



