MR. W. SPOTTISW OODE ON THE CONTACT OF SURFACES. 
261 
viz. a, b, c, f g, h are the six coordinates of the line of intersection of the planes ot, ot', 
or the axis as it may be termed; and A=0, B=Q, C=0, D = 0, Au’+B^-i-Cb-l-D^Q 
represent five planes passing through the axis, the azimuth of the last plane about the 
axis being determined by the quantity ns ’ : ■&. 
This being so, the quantities a, b, . . A, B, . . will he found to satisfy the following 
relations : — • 
BA-C^+D«=0, 
C/- A/i+D5 = 0, 
A,/-B/+Dc=0, ■ 
A a + B5 -\- Cc = 0, 
i 
afA-bg -\-ch = 0 ; j 
( 4 ) 
while for every point of the plane Au’-f By-cGs-j-D^Q we shall have also 
Again, let 
Then if we write 
bz—cy—ft— A, 
cx—az—gt= B, 
ay—bx—ht— C, ■ 
fx -\-gy+hz=~D, 
Au’-f-By-i-Cb-j-B^O. j 
□ V= 
« , 7 » & > 
a', /3', /, y, 
u , v , W, , 
Bj;? B^, B~, Bf, 
Y, 
(4«) 
(5) 
^(column !)-{-?/ (column 2)+^(column 3)+£(column 4) 
for each of the columns in succession of the expression for □ V, it will be found that 
the following transformation may be effected, viz. 
Again, let 
> 
□ 
> 
□ 
*S 
zUY, t DV= 
A, 
B, 
c, 
D, 
u , 
v , 
IV , 
& , 
B* 
B„ 
Bj;9 B;, 
K K, 
Ki K 0 6 = 
u 
, V , 
w. 
£ , 
2 n2 
B z , 
B fc 
( 6 ) 
( 7 ) 
