210 THE THEORY OF SCREWS. [209 



The centre of a conic of the system has for co-ordinates 



sin A sin B sin G 



sin 2A -p sin 25 -p sin 20 -p 



The locus of the centres of the system of conies is easily seen to pass through 

 the vertices of the triangle of reference. It must also pass through the 

 orthocentre, and hence by a well-known property it must be an equilateral 

 hyperbola. It can also be easily shown that this hyperbola must pass 

 through the &quot; symmedian&quot; point of the triangle, i.e. through the centre of 

 gravity of three particles at the vertices of the triangle when the mass of each 

 particle is proportional to the square of the sine of the corresponding angle. 



In Fig. 40 a system of pitch conies has been shown drawn to scale. The 

 sides of the fundamental triangle are represented by the numbers 117, 

 189, 244 respectively. The equation to the system of conies, expressed in 

 Cartesian co-ordinates for convenience of calculation, is 



= .- 1864116 - + y- ( 2-2135882 - - 6852041 ) 



V p i \ p J 



-1649571^ - 9 46700z - 

 25199-54 + * 



P 



Among critical conies of the system we may mention : 



1. The two parabolas for which the pitches are respectively 



8766 and &quot;3089. 



2. The three cases in which the conic breaks up into a pair of straight 

 lines for the pitches sin 2 A, sin 25, sin 2(7, respectively. Of these, the first 

 alone is a real pair corresponding to the pitch 8256034. The equations of 

 these lines are 



x = 7 84y - 978, 



For convenience in laying down the curves the current co-ordinates on 

 each conic are expressed by means of an auxiliary angle ; thus, for example, 

 in computing points on the hyperbola with the pitch 748984 I used the 

 equations 



x = 66 + 26-1 sec 6 + 132 tan 6, 



y = 161-5 + 40-7 sec 0. 



The ellipse with pitch 9 was constructed from the equations 

 x = - 367 - 168 cos 351 sin 0, 



