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205] PLANE REPRESENTATION OF THE THIRD ORDER. 205
cosines are proportional to 0it 02, 03; for if we take the point infinitely
distant we find that the equations reduce to
x _ y _ z
02 03 ’
Accordingly, the line drawn parallel to the screw through the origin has its
direction cosines proportional to 0it 02, 03, and hence the actual direction
cosines are
0-x_____________0, ' 03
7(9/+ (9? + 033’ Vø12+é»22+^’ y/0x"+02-+ey
The cosine of the angle between two screws, 0 and (f>, will therefore be
Oi<t>x + 0-x<l>2 + 0-^3
7ø?+022+0? + + '
By the aid of the conic of infinite pitch we can give to this a geometrical
interpretation.
The co-ordinates of a screw on the straight line joining 0 and <f> will be
0i+X</)j, 02 + \<f>2> O'.i +
If we substitute this in the equation to the conic of infinite pitch we obtain
02 + 0/ + 03s t- 2X (fl,«/., + 0^2 + 0&) + X2 (</>/ + + </>32) = °-
Writing this in the form
ttX2 4" 2&X + c = 0,
of which Xj and X2 are the roots, we have, as the four values of X, corre-
sponding, respectively, to the points 0 and </>, and to the points in which
their chord cuts the conic of infinite pitch,
Xi, X,, 0, 00.
The anharmonic ratio is
X2’
or
b — \Zb2 — ac
b + ^b2 — ac
If co be the angle between the two screws, 0 and (f>, then
b
cos to = ,
Vac
and the anharmonic ratio reduces to
q—2i(ø
whence we deduce the following theorem :—