PEOFESSOE A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
An Agrjregatc for the Loiresf Order.=t of Derivatives. 
39. It may be remarked (and it is easy to verify the statement) that, if we desire 
an algebraically comjdete aggregate of invariants, involving derivatives of (f) alone up 
to order 2 at tlie utmost, and derivatives of E, F, G up to order 1, and the fundamental 
magnitudes of the first three orders and no other quantities, such an aggregate is 
composed of 
a:.-, ivf 
pi 
V3 ■ ye 
H(y/'g) I (u’.i, ?r'b) 
^-4 ’ yu yi. 
H(u>3) 
y3’ y4 
ye 
or 
yc y4’ ys ’ 
and 
J (?Co, 
Every other invailant of the surface involving only the same quantities that occur 
in these invariants can be expressed algebraically in terms of the members of this 
aggregate. The geometrical significance of each of the members has been obtained ; 
if, therefore, the geometrical significance of any additional invariant is known, the 
algebraic equation expressing the invariant in terms of the retained anoTeo'ate will 
express a property of the surface and the curve. Such additional invariants are 
f/Iv 
provided by and ; they should accordingly lead to properties of the surface 
and the curve. 
40, We have 
Two iTen' lielations. 
.. d K 
7/.; = '^'>1 + + LS) (- ' kv .) ■ 
But 
and 
(LQ - MP)' = (N? - 2MQ + LPt) LP - (LN - 1\P) P- - 1;^ (PP - Q2), 
LQ - MP = A {L (KQ - FP) - P (EJ[ - FL)}; 
hence, taking account i)f the iclatlon among the leading terms of the various 
concomitants, we have 
gu (».„ - »vJ («.„ ./,)}= ^ .IK _ 
ils 
'/] I «,)-»■,01 («- 3 ). 
