of Edinburgh, Session 18 82-83. 
163 
or if we represent by C the known quantity, 
C= + 
we get for the locus of 
(38) St^P. + C^O. 
This represents a plane perpendicular to g, its distance from the 
origin 0 of the vectors p being = ?yC. 
The value of dp^ becomes, by (36) and (37), 
dp, = dp^ + rjSrjdp^ , 
or, 
dp, = Y{Yyjdp-^.7j) , 
namely, a vector perpendicular to -q, as we intended it to be, and 
known, g and dp^ being given. 
If we subtract (36) member to member from (34) we get now 
dp = dp, + dr , 
where we put 
dr = 2r]Sr](p — p,) , 
The displacement dp is thus represented by two components at right 
angles to each other : dp, being constant for any point p, and dr 
representing the double of the distance of the extremity of p from 
the plane (38) in which the extremity of p, is situated. 
As p, is arbitrary in this plane, we may take rjC in its place, so 
that 
dr = 2rjSr)(p - -qC) = 27](f>'qp + C) . 
If BM be the projection of p - ?;C on a plane passing through O 
and containing the direction rj, and if represents in the 
plane of projection the intersection with the plane (38), then 
MN == rj^r]{p - rjC) and dr = 2MN = MM'. 
The points M and M' at both extremities of dr are therefore sym- 
