]‘j;oi'i:ss(ji; A, i;, iajksvtii ox the 
324 
ciiid we have already (§ 3) Heen that 
I _ (-O //, -) 
c [u, v, w) 
The invariants, wliich involve the magnitudes a, h, c, j\ (), A but none of their 
derivatives, and which involve derivatives of <^, <j)', of the first order, may be called 
the differential invariants of the first order/''" They are the (quantities denoted (§ 19 ) 
by 0^, D0^, D'0^, DDd-)^, 14 -0^, T. In order to give the geometric significance 
of these invariants, it is necessary to take account of the various directions, through 
the point a/ ?/, .-r, as determined ])y the three surfaces (f)', (fj'\ 
It wall be convenient (mainly for the sake of la-evity) to ado})t an alternative 
notationt for derivatives of a, y, z with regard to a, r, iv. We write 
(.1 _ 
( a (y ^ 
(X _ 
( 0 
r.B 
CHO 
and so for derivatives of y and 5:. Also, followang C'aylev, we take 
Vi 
, y., y. 
n Co Cx C 35 
to be the minors 
of a^ 
•A. !/i 
> Ux Vx ~ 
15 A’ k 
, respectively in 
a,, 
Vx 
Vx 
.a.. 
0 
Hv, 
We then have 
■'1' 
a = 
.1 
■'T 
+ -'lb 
f 
= x.x.j^ + //.,//.. + ty.,, 
h = 
ay 
+ yd 
U 
= aval + ^yj^i + tyi, 
c = 
+ Vd 
+ 
h 
= aia^ + yyj.2 + 
A = 
+ 
+ ' 
F 
= CACi + A 1 A 3 + CA:;, 
+ vd 
+ CA 
(1 
= C'aM + ViVi + C 3 C 0 
(J = 
H- As" 
+ 
H 
= + ViVz H- CiCo 
Let da denote an element of distance through the qaoiiit, in a direction normal to 
the surface (f) = constant, and let 
<f) {a, c, w) = (a, y, z). 
* The magnitudes a, b, i\ f\ rf, h, by their definition, involve derivatives of j, v, z of the first order, 
t This is Cayley’s notation ; see note to § 1. 
f '.r 
car' 
a'li, 
C'.C 
da VC 
^ 1 
(CX 
va vtv 
a’l^. 
v"X 
0 0~ 
A 
C'X 
CV cw 
C'X 
.) 
CIV^ 
J . 
y.'S’ 
