92
ON THE SIX COORDINATES OF A LINE.
[435
question, “ Given any system of two, three, four, five or six lines considered as belonging
to a solid body, to determine the relations between these lines in order that there
may exist along them forces which are in equilibriumbut for greater clearness I
will consider the several cases in order; it is hardly necessary to remark that when
the forces exist the equilibrium will depend on the ratios only, and that the absolute
magnitude of any one of the forces may be assumed at pleasure.
65. The condition in the case of two lines is of course that these shall coincide
together, or form one and the same line; and the forces are then equal and opposite
forces.
66. In the case of three lines, these must meet in a point and lie in a plane;
and the force along each line must then be as the sine of the angle between the
other two lines.
67. Supposing that the forces are X along the line (a, b, c, f g, h), \ along the
line (a 1} b ly c 1} f 1} g lf Ih), and X 2 along the line (a 2 , b. 2 , c 2 , / 2 , g 2 , A 2 ), the conditions of
equilibrium are \a +A 1 a 1 -f X 2 a 2 = 0, Xh + \ 1 h 1 + \Ji 2 = 0, any two of which determine
the ratios X : \ : X 2 ; these ratios were not worked out ante No. 38 for the reason
that with the coordinates there made use of, a symmetrical solution was not obtainable;
but in the present case, selecting the last three equations, these are
X cos a + Xj cos + X„ cos a 2 = 0,
X cos /3 + X 2 cos & + X 2 cos /3 2 = 0,
X cos y + Xi cos 7j + X. 2 cos y 2 = 0,
giving in the first instance an equation which expresses that the three lines (assumed
to meet in a point) lie in the same plane: and then if 01, 02, 12 be the angles
between the pairs of lines respectively, giving by an easy transformation
X + X x cos 01 + X 2 cos 02 = 0,
XcoslO + Xi + X 2 cos 12 = 0,
X cos 20 + X x cos 21 + X 2 = 0.
68. Putting for shortness A, B, G in the place of 12, 20, 01 respectively, we
thence find
1 , cos C , cos B =0,
cos G , 1 , cos A
cos B , cos A , 1
= sin A
which is the required formula.
: sin B