399 
of Edinburgh, Session 1879-80. 
the rays of single resultant, and are the twofold relation which 
determine a congruency with which they are identical. 
A discussion is given of the complex determined by the relation 
P 2 = /v + gW + • • 0>) 
of which (B) is a particular case. 
The equations to Pliicker’s complex cone and equatorial and 
meridian surfaces are given, and various loci connected with the 
complex are discussed. 
A method of exploring the complex by means of central radii is 
then given. 
It is found that the stretch on any radius that is intersected by 
rays of the complex perpendicular to that radius is in general 
finite. 
An equation for the distances of the ends of this stretch from 
the origin is found, and expressions for the direction cosines given 
for the extreme rays which are at right angles to one another. 
Various results concerning the lengths of perpendiculars are 
given; among them that the sum of the squares of the perpen- 
diculars on three rays mutually at right angles to each other is 
constant. 
The solid locus of the feet of the perpendiculars on the central 
axis generally is found to be the space between the sheets of the 
surface 
x 2 y 2 z 2 
r 2 _y2 + r 2 _ gl + r 2 _ ft 2 = 0 . . (E) 
which is the reciprocal of the wave surface. 
Lastly, the congruency of rays determined by (D), and the 
additional relation 
p 4 + f 2 i 2 + gW + h*p = / v + 9W + ft V (F) 
is discussed, and shown to be of the fourth order. Minding’s 
theorem is shown to hold when f= 0. (It is not true when 0). 
The equation to the surface locus of the feet of the perpendiculars 
on the rays of resultant is found, and so far as mere inspection goes 
is of the twelfth degree. In conclusion, the equations of various 
other loci connected with the congruency are given, or indicated to 
show the power of the methods employed. 
