498 
[370 
370. 
ON THE SIGNIFICATION OF AN ELEMENTARY FORMULA OF 
SOLID GEOMETRY. 
[From the Philosophical Magazine, vol. xxx. (1865), pp. 413, 414.] 
i 
The expression for the perpendicular distance of a point (x, y, z) from a line 
through the origin inclined at the angles (a, ¡3, y) to the three axes respectively, is 
p 2 — x 2 + y 2 -f z 2 — (x cos a + y cos /3 + z cos y) 2 
= (y cos y — z cos ¡3) 2 
+ (z cos a — x cos y) 2 
4- (x cos ¡3 — y cos a. ) 2 ; 
and the remark in reference to it is that, if at the given point P we draw, perpen 
dicular to the plane through P and the given line, a distance PK equal to the 
distance of P from the given line, then the expressions 
y cos y — z cos ¡3, z cos a. — x cos y, x cos ¡3 — y cos a, 
which enter into the preceding formula, denote respectively the coordinates of the point 
K referred to P as origin. 
If the given line instead of passing through the origin pass through the point 
x 0 , y<>, z 0 , then the corresponding expressions are of course 
(y — yo) cos y — (z — z 0 ) cos /3, (z — z 0 ) cos a — (x — x 0 ) cos y, (x — x 0 ) cos ¡3 — (y — y 0 ) cos y, 
and if we denote the “ six coordinates ” of the given line, viz. 
cos a, cos ¡3, cos y, y 0 cos y — z 0 cos /8, z 0 cos a — x 0 cos y, x 0 cos /3 — y 0 cos y, 
b y 
a , b , c , f , g , h