A Treatise on the Theory of Screws

282 THE THEORY OF SCREWS. [267, and we learn that this expression will remain absolutely unaltered provided that we only change from one set of co-reciprocals to another. In this / is perfectly arbitrary. 268. Property of the Pitches of Six Co-reciprocals. We may here introduce an important property of the pitches of a set of co-reciprocal screws selected from a screw system. There is one screw on a cylindroid of which the pitch is a maximum, and another screw of which the pitch is a minimum. These screws are parallel to the principal axes of the pitch conic (§ 18). Belonging to a screw system of the third order we have, in like manner, three screws of maximum or minimum pitch, which lie along the three principal axes of the pitch quadric (§ 173). The general question, therefore, arises, as to whether it is always possible to select from a screw system of the nth order a certain number of screws of maximum or minimum pitch. Let ... 06 be the six co-ordinates of a screw referred to n co-reciprocal screws belonging to the given screw system. Then the function or P1Ø1 + ... + p^> is to be a maximum, while, at the same time, the co-ordinates satisfy the condition (§ 35) S0? + 220A cos (12) = 1, which for brevity we denote as heretofore by jR = l. Applying the ordinary rules for maxima and minima, we deduce the six equations From these six equations 0lt... can be eliminated, and we obtain the determinantal equation which, by writing x = 1 +pä, becomes 1—xpi, cos (21), cos (31), cos (41), cos (51), cos (61) cos (12), 1 — xp2, cos (32), cos (42), cos (52), cos (62) cos (13), cos (23), 1 — xps, cos (43), cos (53), cos (63) cos (14), cos (24), cos (34), 1 - xpi, cos (54), cos (64) cos (15), cos (25), cos (35), cos (45), 1 — xps, cos (65) cos (16), cos (26), cos (36), cos (46), cos (56), 1 — xps