A Treatise on the Theory of Screws

 286 THE THEORY OF SCREWS. [270- Equating these two values of the pitch we ought to have for every screw $ + m202 4- m2d2 + ?i62</>83 = »x2</>is + n32</>»3 + n8a</>3a + n,2</>42 + mfd? + mA62. But it can easily be seen that this equation is impossible. Let H be the screw to which all the screws of the five-system are re- ciprocal, and let us choose for S the screw reciprocal to A1; A2, As, B5, H. The fact that >8 is reciprocal to H is of course implied in the assumption that S belongs to the five-system, while the fact that S is reciprocal to each of the screws Alt A2. A:l, B5 gives us ^ = 0, 02 = O, 0S = O, </>6 = 0. Hence we would have the equation n^i2 + n./<£,2 + n32</>33 + r,43</>42 + = 0, which would require that all the co-ordinates were zero, which is im- possible. In like manner any other supposition inconsistent with the theorem of this article would be shown to lead to an absurdity. The theorem is there- fore proved. We can hence easily deduce the important theorem that three of the screws in a complete co-reciprocal system of six must have positive pitch and three must have negative pitch*. For in the canonical system of co-reciprocalsj the pitches are + a, — a, + b, — b, + c, — C, i.e. three are positive and three are negative, and as in this case the n-system being the six-system includes every screw in space we see that of any six co-reciprocals three of the pitches must be positive and three must be negative. 271. Identical Formulae in a Co-reciprocal System. Let any screw a be inclined at angles al, a2,... a6 to the respective six screws of a co-reciprocal system. Then we have for the co-ordinate an +pn) cos al - dalsin al otn — ' . * This interesting theorem was communicated to me by Klein, who had proved it as a property of the parameters of “six fundamental complexes in involution” (Math. Ann. Band. i. p. 204).