A Treatise on the Theory of Screws

280 THE THEORY OF SCREWS. [266, Let the angle A Oh be denoted by A, the angle A Oy by B, and the angle AOX by </>. To find the component we must decompose X', a twist on X, into two components, one on y, the other on the first screw of reference. The component on y can be resolved along the other five screws of reference, since the six form one system with a common reciprocal. If we denote by y' the component on y, we then have X Xj y sin (6 — J.) sin (</> - B) ~ sin (</> — A) ’ and it a and b be the pitches of the two principal screws on the cylindroid, we have for the pitch of X the equation p = a cos3 </> + b sin2 </>; , dp dp d<b . ,, a, , . , also dX ~ d<j> dX ’ ®ecause enect of a change m is to move the screw along this cylindroid. We have ___ , sin (</> - -B) 1 V sin (</> - A) ’ and as the other co-ordinates are to be left unchanged, it is necessary that y' be constant, so that dX} _ , sin (B — A) d</> sin2(</> — A) ’ and hence dp . sin2 (<b — A) = (b-a) sin 2<A • dX7 v ^ysm(B- A) Also dX' dX' dd> . dX,“ dj> ' dX,= ~ C0S ~ A Hence, substituting in the equation , dp dX' X3x1+2pd^‘°‘ we deduce a = b tan </> tan A : but this is the condition that X and the first screw of reference shall be reciprocal (§ 40). 267. Analogy to Orthogonal Transformation. Ihe emanants of the second degree are represented by the equation (Q i i o V ( d d \2 „ \ßl d* + •'' + & daj f~ dX, + • • • + dxj F when 1 is the function into which f becomes transformed when the co- ordinates are changed from one set of screws of reference to another. If we take for / either of the functions already considered, these equations