A Treatise on the Theory of Screws

267] EMANANTS AND PITCH INVARIANTS. 281 reduce to an identity; but retaining f in its general form, we can deduce some results of very considerable interest. The discussion which now follows was suggested by the reasoning employed by Professor W. S. Burnside* in the theory of orthogonal transformations. Let us suppose that we transform the function f from one set of co- reciprocal screws of reference to another system. Let plt ... ps be the pitches of the first set, and qx, ... q6 be those of the second set. Then we must have Pißi2 + ... + p6/3„2 = q1p12 + ... + qepj, for each merely denotes the pitch of the Dyname multiplied into the square of its intensity. Multiply this equation by any arbitrary factor x and add it to the preceding, and we have (& a+...+ß6 y+x +...+ M ( d cZ \2 . , „ o\ = + ’ ’ ’ + dxj /+ ® + • • • + &/V). Regarding ßlt ... ßf, as variables, the first member of this equation equated to zero would denote a certain screw system of the second degree. If that system were “ central ” it would possess a certain screw to which the polars of all other screws would be reciprocal, and its discriminant would vanish; but the screw ß being absolutely the same as p, it is plain that the discriminant of the second side must in such case also vanish. We thus see that the ratios of the coefficients of the various powers of x in the following well-known form of determinant must remain unchanged when one co-reciprocal set of screws is exchanged for another. In writing the d2f determinant we put 12 for , &c. c/otj > da» 11 + xplt 12 13 14 15 , 16 = 0. 21 '22 + xp2, 23 24 25 26 31 32 33 + xp3, 34 35 36 41 42 43 44 + xpit 45 46 51 52 , 53 54 55 + xp6, 56 61 62 63 64 65 , 66 + xpe Take for instance the coefficient of x5 divided by that of x2, which is easily seen to be 1 É£ + + 1 &£_■ Pi' deci1 p6' da62 ’ Williamson, Differential Calculus, p. 412.