310] THE GEOMETRICAL THEORY. 335 



about the several screws of the system of the third order, a geometrical 

 construction could be obtained for determining the point corresponding to 

 any instantaneous screw, when the point corresponding to the appropriate 

 impulsive screw was known. 



We have now to introduce the simplification of the problem, which 

 results when three of the principal screws of inertia of the body belong 

 to the system. But a word of caution, against a possible misunderstanding, 

 is first necessary. It is of course a fundamental principle, that when a 

 rigid system has freedom of the ?ith order, there will always be, in the 

 system of screws expressing that freedom, n screws such that an im 

 pulsive wrench administered on any one of those screws will immediately 

 make the body begin to move by twisting about the same screw. These 

 are the n principal screws of inertia. 



But in the case immediately under consideration the rigid body is sup 

 posed to be free, and it has, therefore, six principal screws of inertia. The 

 system of the third order, at present before us, is one which contains three 

 of these principal screws of inertia of the free body. Such a system of 

 screws possesses the property, that an impulsive wrench on any screw 

 belonging to it will set the body twisting about a screw which also belongs 

 to the same system. This is the case even though, in the total absence of 

 constraints, there is no kinematical difficulty about the body twisting about 

 any screw whatever. 



As there are no constraints, we know that each instantaneous screw, of 

 zero pitch, must be at right angles to the corresponding impulsive screw 

 ( 301). This condition will enable us to adjust the particular homography 

 in the plane wherein each pair of correspondents represents an impulsive 

 screw and the appropriate instantaneous screw. 



The conic, which is the locus of points corresponding to the screws of a 

 given pitch p, has as its equation ( 204) 



3 *-p(6? + 6* + 6&amp;gt; 3 2 ) = 0. 



The families of conies corresponding to the various values of p have a 

 common self-conjugate triangle. The vertices of that triangle correspond to 

 the three principal screws of inertia. 



The three points just found are the double points of the homography 

 which correlate the points representing the impulsive screws with those 

 representing the instantaneous screws. Let us take the two conies of the 

 system, corresponding to p = and p = oo . They are 



3 2 = ........................... (i), 



3 2 = ........................... (ii). 



