300 BROOKS — ORTHIC CURVES. [May 20, 



intersection of the hands, has a constant orientation from jS and y, 

 and in fact generates the orthic curve of the second order given by 



(^ — /5) (x — r) =p'^', 



which is the hyperbola required. 



V. Mechanical Generation of an Orthic Cubic. 



A mechanism which will actually draw an orthic cubic is very 

 much to be desired. One might be made in some such way as the 

 following : Suppose three hands like those described above (IV) to 

 be pivoted at a, /5 and y. Let them be held together in such a way 

 that, while each is free to move along the others, they must always 

 meet in a point, which is to be the tracing point. Each hand is to 

 receive its motion from a cord wound about a bobbin on its axle. 

 The bobbins are to be equal in diameter. The cords pass through 

 conveniently placed pulleys, and are kept tight and vertical by 

 small equal weights at their ends. Consider, to fix ideas, those 

 three weights which by their descent give the hands positive 

 rotation. If, now, the tracing point be moved along an orthic 

 cubic which has a, /?, y for a triad, the total turning of the bobbins 

 will be zero, and as a consequence the total descent of the weights 

 will be zero. Conversely, if we can move these vertically and in 

 such a way that the total descent will be zero, the tracing point can 

 move only along an orthic cubic. This result will be obtained if 

 the centre of gravity of the three weights can be kept fixed. It will 

 not do, however, to connect the three weights by a rigid triangle 

 pivoted at its centre of gravity, for then they will not move ver- 

 tically. But since a parallel projection does not alter the centroid 

 of a set of points, the desired result will be attained if the weights 

 are constrained to vertical motion by guides of some kind, and are 

 kept in a plane which always passes through the centre of gravity of 

 one position of the weights. 



VI. The Orthic Cubic through Six Points of a Circle. 

 Consider the general orthic cubic given by 



x^ — ^0-^"' + ^1"^ — <^2 -|- (^z'^ — (^i^^ -\- a5X^\= o. 

 It cuts the unit circle, 



XX = 1, 



