488 THE THEORY OF SCREWS. 



which possess the required property. Choose any screw a and then take any screw 

 ft whose co-ordinates satisfy these two conditions. 



We shall also use the plane representation of the 3-system. 

 Let U = be the pitch conic. 



V = be the imaginary ellipse obtained by equating to zero the expression for 

 the Kinetic Energy. 



Then the vertices of a common conjugate triangle are of course the principal 

 Screws of Inertia and generally there is only one such triangle. 



It may however happen that U and V have more than a single common 

 conjugate triangle, for let the cartesian co-ordinates of the four intersections of U 

 and V be represented by 



X lt 111. ) X 2) 2/2J X 3) 2/3J a 4&amp;gt; 2/4 



As all the points on V are imaginary at least one co-ordinate of each intersection 

 is imaginary. Suppose y l to be imaginary then it must be conjugate to y. 2 . If 

 therefore the conic U touches V y l and y. A must be respectively equal to y% and y 4 . 

 Hence we have only two values of y, and these are conjugate. Substituting these 

 in U and V we see that there can only be two values of x, and consequently the 

 intersections reduce to two pairs of coincident points. 



Hence we see that V cannot touch U unless the two conies have double contact. 



In this case the chord of contact possesses the property that each point on it is 

 a principal Screw of Inertia while the pole of the chord with respect to either 

 conic is also a principal Screw of Inertia. 



If U and V coincided then every screw of the 3-system would be a principal 

 Screw of Inertia. 



The general theory on the subject is as follows. 



Let (7= be the quadratic relation among the co-ordinates of an ?t-system 

 which expresses that its pitch is zero. 



Let V = be the quadratic relation among the co-ordinates of a screw if a body 

 twisting about that screw has zero kinetic energy. 



The discriminant of S = U + A V equated to zero gives n real roots for A. These 

 roots substituted in the differential coefficients of S equated to zero give the 

 corresponding principal Screws of Inertia. If however there be two equal roots 

 for X then for these roots every first minor of the discriminant vanishes. In this 

 case S can be expressed as a function of n - 2 linear quantities. Perhaps the most 

 explicit manner of doing this is as follows. 



Let S = a u 6f + a^O* + 2a,AO, + . . . + a nn O n \ 



and let ,-!^ -I !*? 



