Linkages

Crossed Parallelogram.

After Sylvester's lecture, a period arose in England during which much attention was paid to the subject of linkages, notably by Sir A. B. Kempe and H. Hart.

Tchebichoff had proved to his own satisfaction that no linkage of five bars could accurately con vert circular motion into recti linear motion; but in 1877 Hart produced one (Proc. Lond. Math.

Soc.), thus showing the danger of trying to prove a negative. He had also previously invented one of four links (Messenger of Mathematics, 1875). This mech anism consisted of a crossed parallelogram EAND (fig. 2), formed by rotating the triangle ACD about the diagonal AD. If any straight line KF be drawn parallel to AD or EN, cutting the links AN and ED in P and Q, then KP•KQ is a constant ; so that if K is fixed, the points P and Q describe inverse curves. If the link AE is fixed and K is the mid point of AE, F will describe the inverse of a conic; and if we add a Peaucellier cell KLMHF, M will describe the conic. If ED:EA as 1 : V 2, F describes a lemniscate and M a rectangular hyperbola.

Other Linkages.

Linkages can thus be made to describe other curves. One of the most interesting is that devised by G. H. Darwin (Proc. Lond. Math. Soc., vol. vi.) for the purpose of drawing the equipotential curves of a system of electrically charged points. If we take two equal Peaucellier cells and pivot their two K points together, and pivot the F point of one and the M point of the other at a fixed distance apart, and ensure (as can be done mechanically in several ways) that the two cells are always equally deformed, then K describes a Cassinian oval with foci at the fixed pivots, for then the product of the focal dis tance of a point on the curve is constant.

M. Saint Louis showed how the solution of a numerical cubic equation can very simply be found by a three-bar linkage (Comptes Rendus, 1874), and as pointed out by Kempe (Proc. Lond. Math. Soc., vol. vii.), an algebraic equation of any degree with numerical coefficients can be solved by linkages of many bars.

The bars AC, AN, DC, DN, DE (fig. 2), with fixed pivots at A and E forms what is called a kite linkage. A great impetus was given to the subject of the three-bar linkages by Darboux (Bull. des Sci. Mathemat., 2° series, tome iii.). He shows that

if a number of kite linkages revolving about the same fixed points are joined together so that the AN of the second coincides with the AC of the first, and so on, and that if the last AC link happens to coincide with the first AN, it will do so in any state of deformation of the compound linkage. The linkage is then said to "close." In his demonstration he employs elliptic functions, the application of which to three-bar motion, originally suggested by Cayley (Phil. Trans. Roy. Soc., 1861) greatly facilitates the solu tion of problems of closure. Fig.

3 gives a very simple form of closed kite linkages and fig. 4 is a photographic reproduction of one derived from a combination of the innumerable vector formulae which arise from the theorems given by Col. Hippisley in Proc. Lond. Math. Soc., series 2, vol. xiii. In both these linkages all the bars will revolve through 360° when the linkages are deformed, though they both appear to be rigid. (See MATHEMATICAL MODELS.) (R. L. H.) LINKOPING, a city of Sweden, the seat of a bishop, and chief town of the district (lan) of ostergdtland. Pop. (1933) 31,005. Linkoping was a bishop's see in 5082 and it was at a council held in the town in 1153 that the payment of Peter's pence was agreed to at the instigation of Nicholas Breakspeare, afterwards Adrian IV. The coronation of Birger Jarlsson Valde mar took place in the cathedral in 1251; and in the reign of Gustavus Vasa several important diets were held in the town. It is situated 142 m. by rail S.W. of Stockholm, and communi cates with Lake Roxen m. to the north) and the Gota and Kinda canals by means of the navigable Stinga. The cathedral is a Romanesque building with a beautiful south portal and a Gothic choir. It contains an altar-piece by Martin Heemskerck (d. In the church of St. Lars are some paintings by Per Horberg (1746-1816), the Swedish peasant artist. Other buildings of note are the episcopal palace (1470 1500), afterwards a royal palace, and the old gymnasium founded by Gustavus Adolphus in 1627, which contains a valuable library of old books and manuscripts and a museum. There is also the ostergotland Museum. The town has manufactures of tobacco, cloth and hosiery.