two Planes, &c. referred to three oblique Co-ordinates. 201 
are situated, one on each of these lines; 5, \ the distances of these 
points from the origin. Then 
D* = & + 8?-—2080 cos. y..... - (2). 
But by equation (1), we have 
D’ = (x#—a)? + (y-y'f + (s—2') + 
+ 2(x—2") (y—y') cos. ry +2(y—y') (—2') cos. y2+2(2—2’) (w—2’) Cos. 22. 
Also 
o= a + y +2°+ Qry cos. vy + 2yz cos. YS + 2H COS. ZL, 
Oo = a? ty” + 2° + Q2'y' cos. ry + 27/2! cos. ¥% + 222’ cos. 2x; 
putting these values in (2) and reducing, 
ve +yy +22'+ (ay +2'y) cos.xry+ (yx +y'z) cos. yz + (za +2'z) cos. zx 
= cos. 97. Vfa°+y?42°+ 2xy cos. vy + 2y2 cos. yX + 22x08. zx}. Vfa4+&c. } 
And putting for 2, y, «’, y’, their values in z, 2’, and dividing by zz’, 
we shall find 
aad +bb'+1+(ab'+a’b) cos. ry +(b+b') cos. yz+ (a+a’) cos. xz 
Ce a V (a+b? +1+2ab cos. ry + 2b cos. yz + 2a cos. xz). V(a* + &e.) ’ 
the second factor of the denominator differing from the first only 
in having a’, b’, instead of a, 6: and thus we have the required 
angle. 
When the co-ordinates are rectangular, cos. xy, cos. yz, cos. xz, 
are each 0, and 
aad +bb+1 
V(ev + b+ 1). V(a* +b” +1)’ 
which is the known formula. 
From the above expression, we may deduce the fundamental 
formula of Spherical Trigonometry. 
Let the two given lines be in the planes xz, yz, respectively, 
and both perpendicular to the axis of z. In this case, their 
angle will measure the inclination of the planes xz, yz, and 
their equations will be 
cos. 4 = 
Vol. II. Part I. Cc 
