536 REPORT— 1881. 



and let B and B' be the conjugate polars of the second with the pitch fi. A pair 

 of common transversals can be drawn across the four lines A A' B B'; these are 

 the common perpendiculars to A and B ; let their lengths be x and y. 

 Then the condition of which we are in search is, 



cos .r cos «/ sin (a + 0) — sin x sin y cos (a — ^) = 0. 



If a system has freedom of the third order, there must be a doubly infinite 

 system of screws about which it can twist; among these let us take three of pitches 

 a, viz. A, A' and B, B', and C C Draw a transversal X across A, B, C-, then a/ 

 will intersect A', B', C. Attribute to x a pitch - a, then x will be neutral to 

 A; B, C. Similarly y and s may be drawn across A, B, C. It is obvious that any 

 screw neutral to x, y, z, must belong to the required three-system. Hence it 

 follows that all the generators of the hyperboloid ABC, and of the same system. 

 as A, B, 0, must belong to the three-system, and thus we have the result that — 



All the, scren'S of given intch belonging to a three-system lie on tivo quadrics 

 lohich are reciprocal imlars with regard to the absolute. Each screw is made up of a 

 generator on one quadric, and its conjugate polar on the othe?-. 



Draw a common tangent plane to the absolute, and the two reciprocal quadrics 

 S and S'. Let T be the point in which this plane meets the absolute. It is obvious 

 T must lie on S and S , and that two rays through T in the tangent plane must be 

 a pair of conjiigate polars forming a screw with the given pitch. Let T L and 

 T M be the two generators of the absolute through T, then any pair of lines through 

 T which formed an harmonic pencil with T L and T M would be a pair of conju- 

 gate polars. The pitch of the screw of the system whose conjugate polars pass 

 through T is therefore ambiguous, and consequently every different pair of 

 hyperboloids for each different pitch must touch this tangent plane and pass 

 through M, T. 



As there can be eight tangent planes to the absolute and a pair of reciprocal 

 quadrics, so it follows that every pair of quadrics must touch these planes and pass 

 through the eight points of contact. We thus have the one degree of freedom left 

 to the quadrics corresponding to the different varieties of pitch. In ordinary space 

 these planes reduce to the four already mentioned. 



13. On a Theorem relating to the Description of Areas. By William 

 WoOLSET Johnson, Professor of Mathematics in the Naval Academy, 

 Annapolis, U.S. 



Let a straight line AB move in any manner in a plane, and let pj and p, 

 represent the distances of the extremities A and B from the point about which the 

 line is for the instant rotating ; then, (p denoting the line's inclination to any fixed 

 line, the area swept over by the line is denoted by 



iP^-p-") # = {Pi±Pi(p.^-p^)d4,. 



In this expression the direction AB is regarded as the positive direction, and 

 thus an area which passes from the left to the right side of AB is positively gene- 

 rated, while an area which passes from the right to the left side of AB is 

 negatively generated. It is easily seen that, with this interpretation, the expression 

 represents the area swept over by AB whatever be the signs of pj and pj ; and 

 consequently if I denotes the length of AB and p,„ the distance of its middle point 

 from the centre of rotation at the instant, the expression for the area is 



lp,„ d(p. 



Now let A and B describe two closed curves (whose areas may be denoted by A 

 and B) returning simultaneously to their original position ; and let the perimeters 

 of these curves be described in the positive direction. Then every point in the 

 area A, and not in Mie area B, will pass at least once from left to right under the 

 line AB, and in all cases once more in that direction than in the opposite direction. 



