XI. On the Angle made by two Planes, or two straight 
Lines, referred to three oblique Co-ordinates. 
CoMMUNICATED By W. WHEWELL, A.M. F.R.S. 
FELLOW AND TUTOR OF TRINITY COLLEGE, 
[Read Nov. 24, 1823.] 
1. Ler there be three co-ordinates, x, y, z, making any angles 
with each other, and let the dihedral angle, made at the axis 
of x by the planes xy and xz, bea; that at the axis of y be 8B, 
and that at the axis of x,y. Let there be two planes referred to 
these co-ordinates, and let their equations be 
Ax+ By+Cz=m, A’x+ By+Cz=m’'; 
it is required to find @, the angle contained by these planes. 
Let three rectangular co-ordinates, having the same origin 
as the others, be assumed, and let 2, y,, z,, be the rectangular 
co-ordinates of the point of which the oblique co-ordinates are 
x,y,%. Let A be the origin, and AM, MN, NP, the rectangular co- 
ordinates, and through P let a plane be drawn parallel to the plane 
of yz, and meeting the axis of x in L: and let planes, parallel 
to this, be drawn also through M and N, meeting 4z in AZ, K. 
Let Al be a line perpendicular to the plane yz, and let the planes 
