Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 7)

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
	        
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