356 THE THEORY OF SCREWS. [325- 



325. A Differential Equation satisfied by the Kinetic Energy. 



If the pitch p of the screw 6 about which a body is twisting receive a 

 small increment 8p while the twist velocity is unaltered the change in 



kinetic energy is 



MpSpv*. 



But the addition of 8p to p has the effect ( 264) of changing each canonical 

 co-ordinate 0^ into 



The variation thus arising in the kinetic energy equated to that already 

 found gives the following differential equation which must be satisfied by uf, 



a (Of - 0f) + b (6? - 0f) + c (0? - Oft 



_ dul\ 03 + 04 fduf _ du 



G fduf _ duf\ 

 \d0 5 d0J 



_ 

 d0 1 d0 2b d0 3 d0 2c 



If we assume that u e - must be a rational homogeneous function of the 

 second order in 1) ...0 6 we can, by solution of this equation, obtain the 

 value of UQ- given in the last Article. 



326. Co-ordinates of Impulsive Screw in terms of the Instan 

 taneous Screw. 



If a lt ... 6 be the canonical co-ordinates of an instantaneous screw and 

 771, ... r) 6 the corresponding co-ordinates of the impulsive screw, then we have 



( ( J9)&amp;gt; 



1 du a * 1 duf , 



e?/! = - -j , e?; 2 = --- j , &c., 

 a d^ a da 2 



and we obtain the following : 



+ ea^ = (+ pf + a 2 ) ^ (+ p? - a 2 ) a 2 + (az - bz - If) 3 



+ (az, + bz - 1 3 2 ) a 4 + (- ay + cy - I?) a 5 + (- ay, - cy - If) a c 



- eaT;, = (+ pi 2 - a 2 ) ^ (+ pf + a 2 ) 2 + (- az - bz - 1 3 Z ) a 3 



(- az + bz - I/) a 4 + (+ ay + cy - If) 6 + (+ ay, - cy - If) a G 

 + eby 3 = (+ az, - bz, - 1 3 -) a, + (- az - bz Q - If) a 2 + (+ pf + 6 2 ) a 3 



(+ p.? - b 2 ) 4 + (+ bx - cx - I*) a 5 + (6^0 + cx - /, 8 ) 6 



- ebr) 4 = (+ az Q -f bz - 1 3 ) a, + (- az + bz - If) 2 (+ p 2 2 - b 2 ) a 3 



(+ p./ -f b 2 ) 4 -f (- bx - cx - l^) a 5 + (- fo + cx - li 2 ) a ti 



+ ecvjr, = (- ay + cy - If) a a + (+ ay + cy - If) 2 + (+ bx - cx - If} &amp;lt;* 3 

 (- bx - cx - If) a, + (+ pf + c 2 ) 5 (+ pf - c 2 ) 8 



- ecrj e = (- ay, - cy - If) a, + (ay, - cy - If) a 2 + (+ bx + cx - If) 3 



(- bx, + ex, - If) 4 + (+ pf - c 2 ) 5 + (pf + c 2 ) &amp;lt;V 



