Surface locus 
of the feet of 
the perpendi- 
culars.on the 
rays of the 
congruency. 
Perpendiculars 
on the four 
rays having a 
given direction. 
Locus of feet 
of perpendicu 
lars on rays 
parallel to 
530 MINDING’S SYSTEM OF FORCES. 
Hence for the locus of the feet of the perpendicular (&7Z) on a ray of the 
congruency common to the two complexes (8) and (9) we have the equation 
LE {et+ (PE +g +PO)—(G + hp +g} (Gh) ] 
+[n{&. JF + [@{&. |} 
which is apparently of the twelfth degree. 
Through a point at an infinite distance in the direction (/mm) four rays of 
the congruency in general pass. From what has been seen already the perpen- 
diculars on these four are each equal to 
SPP + 9m +n « 
0, (36) 
The locus of the feet of the perpendiculars on the rays of the first complex, 
~ (8) or (33), which are parallel to a given plane is 
fri ye (p° =F?) (nt —nn)' + (p°—g?)(nE— Ib)? + (p? 12) (lq —mé)? =. (37) 
The corresponding locus for the second complex, (9) or (34), is 
{ptt (PE + gy + H°C!) —F 4} (me — nn)’ 
+ &e. + &e. =) (88) 
Perpendiculars The locus of the feet of the perpendiculars on the rays of the common 
toogiven congruency which are parallel to the given plane is the intersection of these two 
plane. 
Perpendiculars 
on rays passin 
through a 
given line, 
surfaces, a curve, therefore, of the twenty-fourth degree. 
: The locus of the feet of perpendiculars on the rays that pass through a 
given line may be found by substituting in (37) and (38) 
U(p’—a—byn—cl) — (E—a) { (E—a) + m(y—8) + u(Z—c)} 
for and so on. 
ml—nNn , 
An endless variety of similar results might easily be given, but we have 
already sufficiently exemplified the fertility of the methods employed. 
: ’ , (Added May 1, 1880.) 
Since the above paper was written, I have seen the second part of the 
German translation of Somorr’s “ Mechanics.” I find there a discussion of 
Minpine’s System of Forces in which Ropricuss’ co-ordinates are used. A 
proof of Minpine’s theorem is also given somewhat resembling the second of 
those given above. Somorr, however, looks at the matter almost entirely 
from the statical point of view, and, as I think that many parts of the foregoing 
paper have still an interest of their own, I have allowed the whole to stand 
with a slight addition on page 9. 
