MINDING’S SYSTEM OF FORCES. 527 
the theorem, obviously true from (8), that, if p, p. p; be the perpendiculars from 
the centre on three rays whose directions are mutually at right angles, then 
Prt pe tpra=f72t+g2+h?=const. . . (22) 
and if p, and p, be the perpendiculars on two rays perpendicular to each other, 
and to a fixed line, then 
p17 + po” =const. 
(Inserted May 1, 1880.) 
[The results of (19), (20), and (22) shew the close analogy of the properties 
of the complex with those of a quadric surface. Taking this point of view, and 
starting from (17) and (18), we are led to many interesting results, among 
which the following is noteworthy. 
The perpendicular rays at any point of the radius (mn) make equal angles 
with the extreme double rays; and, if #0 be the angle they make with the 
extreme ray whose distance from the origin is p, , then 
p” = p,” cos 70 + p,” sin 70, : : : (23) 
which of course contains, as particular cases, some of the results already 
obtained. This formula has an interesting resemblance to Hamitton’s elegant 
relation, connecting the shortest distance from any ray of a congruency to any 
consecutive ray with the shortest distances corresponding to the virtual foci of 
that ray. See Satmon, “ Geometry of Three Dimensions,” 3d edition, p. 567. ] 
As the result of the above discussion we have the following. 
The surface of the fourth degree 
2rt=(g2 +h?) + (+S %)y2+(f2+g%) . . ~— (28) 
is the locus of points at which the two rays perpendicular to the central radius 
are perpendicular to each other. 
The surface 
2 
2 
apt pop t woe = 9, ae gs) | (Oa 
which is the reciprocal of the wave surface, is the locus of points at which the 
two rays perpendicular to the central radius coincide, and the space between 
its two sheets is the solid locus of the feet of perpendiculars on the rays of the 
complex. 
This elegant theorem is due to Tair. 
We might adopt a more general method of exploration by supposing the 
rays through the exploring radius to be parallel to a fixed plane. The results 
VOL. XXIX. PART II. 6 T 
