A Treatise on the Theory of Screws

66 THE theory of screws. [76, the amplitude p of a twist about p which will produce the same effect as the four given twists. We have seen (§ 37) that the twist about any screw a may be resolved in one way into six twists of amplitudes aait ... aa„, on the six screws of reference ; we must therefore have p'Pi = a'a, +ß'/3j + y'y, + 8'8,, p Pl) — a a0 + ß ße + y ye 4- 8'8a, whence p' and p1; ... pl; can be found (§ 35). A similar process will determine the co-ordinates of the resultant of any number of twists, and it follows from § 12 that the resultant of any number of wrenches is to be found by equations of the same form. In ordinary mechanics, the conditions of equilibrium of any number of forces are six, viz. that each of the three forces, and each of the three couples, to which the system is equivalent shall vanish. In the present theory the conditions are likewise six, viz. that the intensity of each of the six wrenches on the screws of reference to which the given system is equivalent shall be zero. Any screw will belong to a system of the nth order if it be reciprocal to 6 — n independent screws; it follows that 6 — n conditions must be fulfilled when n + 1 screws belong to a screw system of the nth order. To determine these conditions we take the case of n = 3, though the process is obviously general. Let a, ß, y, 8 be the four screws, then since twists of amplitudes a, ß', y, 8' neutralise, we must have p' zero and hence the six equations <*'ai + ß'ßi + y'y, + 8'8, = 0, &c. «'«» + ß'ß„ + y'y, + 8'8e = 0; from any four of these equations the quantities a, ß', y', 8' can be eliminated, and the result will be one of the three required conditions. It is noticeable that the 6 — n conditions are often presented in the evanescence of a single function, just as the evanescence of the sine of an angle between a pair of straight lines embodies the two conditions necessary that the direction cosines of the lines coincide. The function is suggested by the following considerations :—-If n + 2 screws belong to a screw system of the (n + l)th order, twists of appropriate amplitudes about the screws neutralise. Ihe amplitude of the twist about any one screw must be pro- portional to a function of the co-ordinates of all the other screws. We thus see that the evanescence of one function must afford all that is necessary for n +1 screws to belong to a screw system of the nth order.* Philosophical Transactions, 1874, p. 23.