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338 SECONDE PARTIE. — CONFÉRENCES ET COMMUNICATIONS. — SECTION III.

(a). A rotation about the [-axis clisplaces the plane

fx + gy •+■ hz = o

in itself about the fixed point (J, A, B, G), and a rotation about the J-axis displaces the plane

ax -+- by + cs = o

in itself about the fixed point (I, — A, — B, — C).

(6). The planes ax -+- by H- cz = o and fx gy -f- hz = o are per-

pendicular to one another. Their line of intersection passes through the origin and meets the two axes (the I and the J axes) at the points (J, A, B, G) and (I, — A, — B, — G) respectively.

(c). The I-axis meets the plane fx -+- gy -f- h z = o perpendicularly at the point (J, A, B, G) and the J-axis meets the plane ax-h-by-y cz = o perpendicularly at the point (I, — A, — B, — C).

(,d). The conditions © = o, ©' — o being satisfîed, b, c,f, g, h and y7, g'j h/, ab', c' are the Plückerian coordinates of straight lines; the former lies in the plane fx -)- gy + hz — o, the latter in the plane ax -f- by + cz = o, and both t.herefore meet the intersection line of these two planes.

(e). A rotation about the I-axis displaces the line («, 6, c,f7 g, h) in itself and a rotation about the J-axis displaces the line (/’, g', h', ab', c') in itself.

(/). The distances from the origin, along the intersection of the planes fx + gy + hz — o, ax + by -ÿ cz — o, to the points (J, A, B, C) and (I, — A, — B, — G) being denoted by OP and OQ respectively and cos being defined by

(in in\

ek -y e k J,

these distances are determined by the formulae :

cos/iOP =

)

and the perpendicular distance between the two axes is

OP 4- OQ = \ k-k.

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