164 



SCIENCE. 



[N. S. Vol. XIX. No. 474. 



of the Crossley reflector suggested the 

 same principle as being suitable for the 

 axes of large telescopes. These bearings 

 are very simple in construction and con- 

 sist of a ring of hardened 'steel rollers 

 around the axis, in the bearing. The roll- 

 ers fit closely about the axis and, therefore, 

 do not require any frame to hold them in 

 their relative positions. There is no loose- 

 ness and the axis revolves with perfect 

 accuracy, yet easily. 



Such bearings would be fully as efficient 

 in the case of a large overhang of the 

 polar-axis as in the ordinary form of 

 mounting. Where the ends of the polar 

 axis are supported on separate piers the 

 bearings can be made self-aligning. 



A Linkage for Describing the Conic Sec- 

 tions by Continuous Motion: J. J. 

 QuiNN, Warren, Pa. , 

 This linkage is the material embodiment 

 of the facts set forth in the following 

 theorem : 



If one vertex of a movable pivoted 

 rhombus be fixed in position, while the 

 opposite vertex is constrained to move in 

 the arc of a circle, the locus of the inter- 

 section of a diagonal (produced) through 

 the other two vertices, with the radius 

 (produced) of the circle in which the ver- 

 tex moves is a conic. 



If the fixed vertex is in the diameter 

 of the circle, and the directing radius 

 finite, the locus is an ellipse. If the direct- 

 ing radius is infinite and the fixed vertex 

 in the diameter, the locus is a parabola. If 

 the directing radius is finite, and the fixed 

 vertex is in the diameter produced, the 

 locus is a hyperbola. Modifications of the 

 essential features of this linkage give rise 

 to many interesting corollaries involving 

 the geometric construction of the conies, 

 their tangents and normals. 



Circles Represented by iJ.^P-\-Lii!^Q-\-MiJ.li 



-{-NS^O: T. R. Running, Ann Arbor, 



Michigan. 



In the equation discussed m is a variable 

 parameter; L, M and N are constants; P, 

 Q, R and S represent circles. The equa- 

 tion itself represents circles for all values 

 of the parameter. Three circles of the sys- 

 tem pass through each point of the plane. 

 The locus of the centers of the system is 

 a cixbic having eight arbitrary constants. 



There will be a circle orthogonal to the 

 system if any one of the circles P, Q, R, S 

 can be derived linearly from the other 

 three. There are six point circles in the 

 system, all lying upon the locus of the cen- 

 ters. Four circles of the system are tan- 

 gent to any one. Eight pairs of tangent 

 circles have a common linear relation con- 

 necting their parameters. 



The envelope of the system is 



18 LMNPQRS — 27 mP^S^ + L^if ^ Q^E'^ 



— ^UNQ'S + M^PE^) = 



which may be written 



where 



^ = i2Q2 — 3PMB, C= 3PB'—LQN8, 

 B = LMQB — 9PNS. 



It is shown that this is the envelope of 



fi^A + fiB+C=0, 



A, B, C being bicircular quarties which 

 are themselves envelopes of systems de- 

 rived from the original circles. 



The envelope of the radical axes of a 

 particular circle and other circles of the 

 system is a conic. This conic may be said 

 to correspond to the particular circle, and 

 there is such a conic corresponding to 

 every circle of the system. The system of 

 circles represented by 



fi'P + Lfi'Q + M/iB + NS=0 



is called the primary system, and the sys- 



