90 
ON THE SIX COORDINATES OF A LINE. 
[435 
In fact assuming for a moment that the axis of rotation passes through the origin, 
then for the point P coordinates x, y, z, the square of the perpendicular distance from 
the axis is 
( . — y cos 7 + z cos /3) 2 
+ ( x cos 7 — z cos a ) 2 
+ (— x cos /3 + y cos a . ) 2 , 
and the expressions which enter into this formula denote as follows; viz. if through 
the point P at right angles to the plane through P and the axis of rotation we 
draw a line PQ, — perpendicular distance of P from the axis of rotation, then the 
coordinates of Q referred to P as origin are 
. — y cos 7 + z cos /3, 
x cos 7 — z cos a, 
— x cos /3 + y cos a . , 
respectively. Hence the foregoing quantities each multiplied by A are the displacements 
of the point P in the directions of the axes, produced by the rotation A. 
60. Suppose that the axis of rotation (instead of passing through the origin) pass 
through the point (x 0 , y 0 , z 0 ); the only difference is that we must in the formula 
write (x — x 0 , y — y 0 , z — z 0 ) in place of (x, y, z): and attending to the significations of 
the six coordinates, it thus appears that the displacements produced by the rotation 
are equal to A into the expressions 
. -hy+gz- a, 
hx . —fz — b, 
-gx+fy • -c, 
respectively; which is the theorem in question. 
61. I say secondly that considering in a solid body the point (x, y, z) situate in 
the line {a, b, c, f g, h), and writing 
a, b, c, f, g, h = z cos ¡3 — y cos 7, x cos 7 — z cos a, y cos a — x cos /3, cos a, cos /8, cos 7, 
then for any infinitesimal motion of the solid body the displacement of the point in 
the direction of the line is 
= ap + bq + cr +fl + gm + hn, 
where p, q, r, l, m., n are constants depending on the infinitesimal motion. 
In fact for any infinitesimal motion of a solid body the displacements of the point 
(x, y, z) are 
Sx = l . +ry — qz, 
8y = m — rx . + pz, 
8z = n + qx — py . ,