A Treatise on the Theory of Screws

210 the theory of screws. [209 The centre of a conic of the system has for co-ordinates sin A sin B sin C sin 2^4 — p’ smlB—p’ sinZC—p' The locus of the centres of the system of conics 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 “ symmedian” 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. IR Fig. 40 a system of pitch conics 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 conics, expressed in Cartesian co-ordinates for convenience of calculation, is n ./»rodTt- ’6852041\ / -6852041\ 0 = x2 -7864115 - ---------- ) + v2 ( 2’2135882 - " ] V P ) 7 k p ) + -1649571^ - 946700« - 430'5179?/ + 25199-54 + 5726'031 = o P Among critical conics of the system we may mention : 1. The two parabolas for which the pitches are respectively ’STOe and ‘3089. 2. The three cases in which the conic breaks up into a pair of straight lines for the pitches sin 2/1, sin ZB, sin 2(7, respectively. Of these, the first alone is a real pair corresponding to the pitch '8256034. The equations of these lines are « = 7'84y- 978, x — — 4'06y + 760. 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 0, y — 161’5 + 40-7 sec 0. The ellipse with pitch '9 was constructed from the equations x = — 367 — 168 cos 0 + 351 sin 0, y — +169 + 51 cos 0.