A Treatise on the Theory of Screws

276 THE THEORY OF SCREWS. [261- The functions thus arising are well known as “ emanants ” in the theory of modern algebra. The cases which we shall consider are those of n = 1 and n = 2. In the former case the emanant may be written a df n df ßx/~+ ...+ß6-f-. daT dae 262. Angle between Two Screws. It will of course be understood that f is perfectly arbitrary, but results of interest may be most reasonably anticipated when f has been chosen with special relevancy to the Dyname itself, as distinguished from the influence due merely to the screws of reference. We shall first take for f the square of the intensity of the Dyname, the expression for which is found (§ 35) to be R = «/ + ... -t- a62+ 2axa2 (12) + where (12) denotes the cosine of the angle between the first and second screws of reference, which are here taken to be perfectly arbitrary. The second group of reference screws we shall take in a special form. They are to be a canonical co-reciprocal system, so that R = (\1 + ^-s)2 + (^3 + \1)2 + (\ + V- Introducing these values, we have, as the first emanant, ta^ + X + a.2ßß (12) — (#i + Ma) (\ + ^2) + (/Ai + fJ-t) (Xs 4- X4) + (/z5 + /x6) (X5 + X6); but in the latter form the expression obviously denotes the cosine of the angle between a and ß where the intensities are both unity; hence, whatever be the screws of reference, we must have for the cosine of the angle between the two screws the result SaiÄ + X («,& + aoÄ) (12). 263. Screws at Right Angles. In general we have the following formula for the cosine of the angle between two Dynames multiplied into the product of their intensities:—■ ,n dR . „ dR , „ dR This expression, equated to zero, gives the condition that the two Dynames be rectangular.