:me. w. spottiswoode on the contact of cueves. 
43 
then a:h \ . k 
da 
db 
dk 
db 
■■'-‘it 
□ 
^ |-§ 
T 
m— 1 
db 
□ . 
(15.) 
The equations (11.) are, however, not aU independent. In the first place, as in (10.) 
any one of the three equations is a consequence of the other two, so in ( 12 .) any one 
column or row is a consequence of the other two. Secondly, if any pair of binary com- 
binations of the Ds, e. g. DjDaV and DgDjV, be developed, they will, with the help of 
(10.) or (3.), be found identical. This reduces the system (12.) to three independent 
conditions, of which 
D^V=0, DAV=0, D^V=0 (16.) 
is a type ; as it should do. These three equations will sutfice to determine the three 
independent constants of V, when it is capable of a 3-pointic contact, and no more, 
with U. 
The actual calculations may be considerably simplified by using, instead of (13.), the 
following system, 
V=0, DV = 0, D^V=0 
(where, as before, D stands for any one of the symbols Di, Dg, D 3 ). Of this system (16.) 
is a consequence. 
By way of example, we have for the ordinary tangent 
Y =ax-\-hy-\-cz=0, 
jyV =dDx-{-bT)y-\-cT)z=^, 
whence 
a \ h : c 
^ y ^ (17.) 
Do; T>y Dz 
—yDz—zDy : zT)x—xT)z : xDy—yT>x 
au . au au au 
au w ^ 
dx ' dy ' dz 
( 18 .) 
The equation of the circle of curvature may be put in the form 
hf'-\-''l[ayz-\-})zx-\-cxy) — ^ ; 
to which are to be added, 
liD r‘^-\- 2[dD yz-\-hJ) zx-j-cD xy) = 0, 
hjyr^ _|_ 2[aiyyz -J- hWzx-\-cWxy) = 0 , 
G 2 
