A Treatise on the Theory of Screws

356 the theory of screws. [325- 325. A Differential Equation satisfied by the Kinetic Energy. If the pitch p of the screw 0 about which a body is twisting receive a small increment tip while the twist velocity 0 is unaltered the change in kinetic energy is MpSpØ2. But the addition of bp to p has the effect (§ 264) of changing each canonical co-ordinate 04 into *+$ The variation thus arising in the kinetic energy equated to that already found gives the following differential equation which must be satisfied by w/, a - 0f + b (0S* - 0/) + c - 0f — _ duf , &3 + &4 Cdue2 _ duf 0&+ 06 (due2 duf 2« W6»! døj 2b \d03 døj+~2i \d05~døj' If we assume that ue2 must be a rational homogeneous function of the second order in 0,,... 06 we can, by solution of this equation, obtain the value of wa2 given in the last Article. 326. Co-ordinates of Impulsive Screw in terms of the Instan- taneous Screw. If a3 be the canonical co-ordinates of an instantaneous screw and Vi> ■■■ Vi> the corresponding co-ordinates of the impulsive screw, then we have (§ 99), 1 duf eV1 = - a da4 and we obtain the following: + eap = (+ pi2 + a2) 1 duf 0 ep2 =------, &c., a da2 a2) a2 + (az,, - bz0 - If a3 + («z0 + bz0 — If a4 + (— ay3 + cya - If a5 + (— ay0 — cy0 - If a6 - eay2 = (+ pf - of a4 (+ pf + a2) a2 + (- az0 — bz0 - If a3 (- az0 + bz0 — If a4 + (+ ay0 + cy„ — If a5 + (+ ay0 — cy0 — If a,; (+ p22 + b2) a3 + ebr/3 = (+ az0 - bz0 - If + (- az0 (+ P? ~ &2) a4 + (+ bæa — cxl} — If as + (bx0 + rø0 — If ae ~ eby4 = (+ az„ + bz„ - If + (- az<1 + bzQ - If a2 (+ p.2 - b2) a3 (+ P22 ■+ &2) a4 + (- bæ0 — cæ0 — If a6 + (- bæ0 + cx0 — If «6 + ecy5 = (— ay0 + cy0 — If a4 + (+ ay3 + cy0 — If a2 + (+ bæ0 — cx0 — If a3 (— bx0 — cx0 — If a4 + (+ p32 4- c2) a5 (+ p32 — f a6 — ecpg — (— ay3 — cy0 — If a4 + (a,y0 — cy0 — If a2 + (+ bæ3 + cæa — If a3 (— bx0 + c«0 — If a4 + (4- p32 — c2) as _|_ (pt