435] 
ON THE SIX COORDINATES OF A LINE. 
89 
o 
values which of course satisfy, as they should do, the relation af+bg + ch = 0. It is 
hardly necessary to remark, that the values of a, b, c are not altered on substituting 
foi x 0) y 0 , z 0 the cooidmates x 0 -\- s cos a, y 0 -)-s cos^S, ^o+scos^y of any other point on 
the line. 
58. Considering any two lines (a, b, c, f g, h), (a,, b u c u f u g u l h ), if we define 
the moment of the two lines to be the product of the perpendicular distance into the 
sine of the inclination of the two lines, then we have, Moment 
= afi + bg L + chi + fa x + gb x + hc 1} 
viz. we have now a quantitative definition of the function of the coordinates previously 
called the moment of the two lines. 
For the demonstration of this formula it is to be remarked, that taking on the 
first line a segment of the length r, the coordinates of its extremities being (x 0 , y 0 , z 0 ) 
and (x 0 + r cos a, y 0 + r cos ¡3, z 0 + r cos 7), 
and on the second line a segment of the length 77 the coordinates of its extremities 
being (x 0 ', y 0 ', z 0 ') and (#„' +rjCosoq, 7/ 0 '-f 77 cos &, V + ^iCOsyj) and joining the extremities 
of these segments so as to form a tetrahedron, the volume of the tetrahedron is 
= \ rr i ( a fi + + cJh +f<h + gbi + hci). 
But the volume of the tetrahedron is also equal to £ of the product of the opposite 
edges into their perpendicular distance into the sine of the inclination of the two 
edges ( x ); that is, it is = ^rr x into the moment of the two lines, and we have thus 
the formula in question. 
Article Nos. 59 to 75. Statical and Kinematical Applications. 
The coordinates (a, b, c, f g, h), as last defined, are peculiarly convenient in 
kinematical and mechanical questions, as will appear from the following investigations. 
59. Using the term rotation to denote an infinitesimal rotation, I say first that 
a rotation A round the line (a, b, c, f, g, h) produces in the point (x, y, z) rigidly 
connected with this line the displacements 
8x = X ( . - hy + gz-a), 
8y = \ ( hx . —fz — b), 
Sz =X(— gx +fy . -c). 
1 I take the opportunity of mentioning a very simple demonstration of this formula : taking the opposite 
edges to be r, r,, their inclination =6, and perpendicular distance =h; the section of the tetrahedron by a 
plane parallel to the two edges at the distances 0, li-z from the two edges respectively is a parallelogram, 
the sides of which are r -( h Z z l and % respectively, and their inclination is =6; the area of the section is 
h h 
therefore Ulsin 0.z(h — z) and the volume of the tetrahedron is = p- sin 6 J q z (h-*) dz, — - 6 ir x h sin 6. The 
same result is however obtained still more simply by drawing a plane through one of the two edges perpen 
dicular to the other edge; the volume is then equal to the sum or the difference of the volumes of two 
tetrahedra standing on a common triangular base; and the required result at once followa. 
C. VII. 12