123] FREEDOM OF THE SECOND ORDER. Ill 



are no real screws of zero pitch. If the pitch conic be a parabola, there is 

 but one screw of zero pitch, and this must be one of the two screws which 

 intersect at right angles. 



123. Equilibrium of a Body with Freedom of the Second Order. 



We shall now consider more fully the conditions under which a body 

 which has freedom of the second order is in equilibrium. The necessary 

 and sufficient condition is, that the forces which act upon the body shall 

 constitute a wrench upon a screw which is reciprocal to the cylindroid which 

 defines the freedom of the body. 



It has been shown ( 23), that the screws which are reciprocal to a cylin 

 droid exist in such profusion, that through every point in space a cone of 

 the second order can be drawn, of which the entire superficies is made up of 

 such screws. We shall now examine the distribution of pitch upon such a 

 cone. 



The pitch of each reciprocal screw is equal in magnitude, and opposite in 

 sign, to the pitches of the two screws of equal pitch, in which it intersects the 

 cylindroid ( 22). Now, the greatest and least pitches of the screws on the 

 cylindroid are p a and p$ ( 18). For the quantity p a cos 2 1 + p ft sin 2 1 is always 

 intermediate between p a cos 2 1 + p a sin 2 1 and pp cos 2 1 + pp sin 2 1. Hence it 

 follows that the generators of the cone which meet the cylindroid in three 

 real points must have pitches intermediate between p a and pp. It is also 

 to be observed that, as only one line can be drawn through the vertex of 

 the cone to intersect any two given screws on the cylindroid, so only one 

 screw of any given pitch can be found on the reciprocal cone. 



One screw can be found upon the reciprocal cone of every pitch from 

 oo to + oo . The line drawn through the vertex parallel to the nodal line 

 is a generator of the cone to which infinite pitch must be assigned. Setting 

 out from this line around the cone the pitch gradually decreases to zero, 

 then becomes negative, and increases to negative infinity, when we reach 

 the line from which we started. We may here notice that when a screw 

 has infinite pitch, we may regard the infinity as either + or indifferently. 

 If we conceive distances marked upon each generator of the cone from the 

 vertex, equal to the pitch of that generator, then the parallel to the nodal 

 line drawn from the vertex forms an asymptote to the curve so traced upon 

 the cone. It is manifest that we must admit the cylindroid to possess 

 imaginary screws, whose pitch is nevertheless real. 



The reciprocal cone drawn from a point to a cylindroid, is decomposed 

 into two planes, when the point lies upon the cylindroid. The first plane 

 is normal to the generator passing through the point. Every line in this 

 plane must, when it receives the proper pitch, be a reciprocal screw. The 



