46 
ME. W. SPOTTISWOODE ON THE CONTACT OF CUEYES. 
Writing for convenience 
, bU .BU BU BU „BU BU 
and putting 
V = {abcfglijxyzy ^ 
the general expressions (13.) become 
0= {ahcfghjxyzf, 
0= {alcfghJXy.vJccyz\ 
0 = {abcfgh'yykgjvy' -\-{(ibGfgh\ □ X □ jM/ □ v^jcgz), 
0 = o{ahcfgli^ □ X □ + [abcfgh\ □ □ ‘^v\xyz), 
Q = o{ahcfgh'^ □ X □ □ vy-\-4L{ahcfgh'^ □ □ ^vy\gjv)-\-{al)cfgh\ □ □ ^vjjcyz), 
whence 
a\h:c:f\g\ h= 
3(nx)'“+4xn"x+^n®x . . 6n[jtjDv+i(iJtjm+vD^g^)+yn^+za^gj . 
3xDx+.'rn"x . . ^[Jb^v-\-v^(^)-\-yo^v-\-z^yJ 
X*-|~iTnX . . ^gjV-\-y\Zlv-\-zmg) 
xX . . ygj-\-zv 
of y^ z^ 2yz 2zx 2xy. 
( 22 .) 
The most convenient method for developing these formulce is to begin with forming the 
15 minors derived from the last two lines ; and by means of them, the 20 minors derived 
from the last three lines ; and so on. It may be noticed that in order to pass to the 
corresponding expressions in D, e. g. Dj, it is only necessary to make j3=0, y=0, and to 
di\ide out the as. This gives 
. . BU BU 
The results of the calculations are as follow : the third minors, formed from the two 
lower lines of (22.), and from the columns the lower constituents of which alone are 
written down, will be 
2 2 2 ^ BU , , BU 
Izx, ‘Ixy, 2yz=2f^^, 2yz, 2zx=iz'^, 
x‘,2yz=x(^y^-z^') 
dy 
, 0 ?", 2zx=x'^'^ x^, 2xy=—x^^, 
f, 2yz=-f^, y\ 2zx=y ( 2 ^-;^^) y\ 2xy=^f^, 
z\ 2yz=z^'^, z\ 2zx=-z‘^, z\ 2^=«(*^-y^). 
