A Treatise on the Theory of Screws

274] EMANANTS AND PITCH INVARIANTS. 289 274. A Pitch Invariant. Let hlt ... h6 be the direction angles of any ray whatever with regard to six co-reciprocal screws of reference, then the function cos A, + ... + a6 cos hs is a pitch invariant. For, if we augment the pitch of a by x, we have to write for an... a6 the expressions a, + cos a, ... cos a(il 2^! % and then U becomes a, cos Äj 4-... 4- a6 cos A6 x /cos a, cos A, cos a8 cos hA 2 \ Pi pe ) but from what we have just proved, the coefficient of x is zero, and hence we see that a, cos hi + ... + a6 cos remains unchanged by any alteration in the pitch of a. If we take three mutually rectangular screws, a, ß, y, then we have the three pitch invariants L = 6i cos ai + ... + 0O cos a6, M = Oi cos bi + ... + 0e cos b6, N = 6i cos Ci + ... + 6e cos c6. It is obvious that any linear function of L, M, N, such as fL + gM+KN, is a pitch invariant. We can further show that this is the most general type of linear pitch invariant. For the conditions under which the general linear function • ^i^i + ... + An0n shall be a pitch invariant are that equations of the type Acosai ! ^6cosa6_0. &c Pi P« shall be satisfied for all possible rays. Though these equations are infinite in number, yet they are only equi- valent to three independent equations; in other words, if these equations are satisfied for three rays, a, b, c, which, for convenience, we may take to be rectangular, then they are satisfied for every ray. B. 19