200 Mr. WHEWELL on the Angle made by 
In the same manner we transform P” + Q?+ R”, and the 
expression becomes 
— cos. 60 = 
A Ae , ‘ 1 1 1 1 1 v 
Ad! BB’, CC’ A'B+AB | _ACHAC 9 BCHBC 
a! d* ex Hie de df efi 
APB. C2, VAB: 2AC sa 1236) + 2A4'B' BAC! 2B'C 
Ji Gto Te e008: a cos. ar cos. «) (Fto+ Te 08 Y= if cos. B— F 
If we suppose a Jie to be Been. with its center at 
the origin, it will cut the planes vy, xz, yz, in three arcs, which 
will form a triangle, whose angles will be a, 6, y. And, if per- 
pendiculars be let fall from its angles on the opposite sides, d, e, f, 
will be the sines of these perpendiculars. 
2. Let there be three oblique co-ordinates as before, and 
two lines of which the equations are respectively 
e=az+a w2=az+a). 
Sa the pt 
it is required to find » the angle contained by these lines*. 
Let cos. xz, cos. vt, &c. indicate the cosines of the angles 
which the directions of the lines x and z, v and f, &c. make with 
each other. 
Now if D be the diagonal of any parallelepiped, ¢, uv, v its sides, 
D = +u> 4+ v°+ 2Quv cos. uv + 2vt cos. vt + 2tu cos. tu...... (1). 
The angles uv, &c. being measured at the solid angles to which 
the diagonal is drawn. 
Let there be lines passing through the origin, parallel to the 
given lines; the equations to these lines will be respectively, 
2 Th = a Ge SSS 
y = be >» an fle . 
Let D be the distance of any two points from each other, which 
* The investigation which follows, is by J. W. Lubbock, Esq. of Trinity College. 
