390] THE THEORY OF PERMANENT SCREWS. 427 



We thus find that when T is referred to the three permanent screws of the 

 system, its expression must be 



T= aB* + 60 2 2 + c0 3 2 + ZfdA + %ff$A + %h0A 



+ (ji-v) 010 A + (v-\) 0M + (X - A*) S A- 



Let 77&quot; be the intensity of any wrench acting on a screw 77 belonging to 

 the system, and let 2-or lr , represent the virtual coefficient between 77 and the 

 first of the three screws of reference. 



Then, substituting for T in Lagrange s equations, we have 

 + ah + h 2 + g 3 -(p-v) eA = -n-^7?&quot;, 



If 77 be the restraining screw, then an appropriate wrench ?/ should be 

 capable of annihilating the acceleration, i. e. of rendering 



0j = ; 2 = ; S = ; 

 whence the position of 77, and the intensity 77&quot; are indicated by the equations 



(V - 



We can now exhibit the nature of the correspondence between 77 and 0, for 



If we make H=$i$A+n&quot;&amp;gt; and omit the dots over lt &c., we have 



+ w 32 77 3 ) - H (a - 7), 

 03 Oia*?! + ^ 32 i?2 +p 3 r ns) =H(@- a). 

 We may reduce them to two homogeneous forms, viz. 



where L = - M=; N = \ 



d^ * ^772 ^773 



390. Geometrical Construction for the Permanent Screws. 



We see that 77 must lie on the polar of the point lt 2 , 3 with respect 

 to the pitch conic ( 201) or the locus of all the screws for which 



