393] THE THEORY OF PERMANENT SCREWS. 431 



The permanent screw on this cylindroid will be one whereof the restrain 

 ing screw coincides with P. In gene.ral, the points corresponding homo- 



Fig. 44. 



graphically to the points on the ray AB will form a ray CD. The inter 

 section 0, regarded as on CD, will be the correspondent of some point X on 

 AB. The restraining screw corresponding to X will therefore lie at P, and 

 will be provided by the constraints. Accordingly, X is a permanent screw 

 on the cylindroid, and it is obvious from the construction that there can be 

 no other screw of the same character. 



We can also deduce the expression for T in the two-system from the 

 expression of the more general type in the three-system ; for we have 



T= T + o - v) o; e A +(v-\) o;d A + (*-/*) # 3 0A. 



Consider any screw on the cylindroid defined by 



substituting, we obtain 



(0, 8 - 2 3 ) [(X - ft) P0 2 + (\-v) Q9 3 ], 



which we already know to be the form of the function in the case of the two- 

 system ( 384). 



393. Freedom of the Fourth Order. 



The permanent screws in the case of a rigid body which has freedom of 

 the fourth order may be investigated in the following manner: If a screw 

 be permanent, the corresponding restraining screw 77 must be provided by the 

 reactions of the constraints. All the reactions in a case of freedom of the 

 fourth order lie on the screws of a cylindroid. On a given cylindroid three 

 possible 77 screws can be found. For, if we substitute a l + \^ 1 , 2 + X/3 2 , &c., 

 for T/!, 7/2, &c., in the equation 



{ c-a [ 



b 7? 3 - 7/4 



