895 
OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
The various indices of these quantities, being the powers of V by which they must 
be divided to become absolute invariants, are :— 
Index 2, I v/,) ; 
Index 3, J {tv., iv'.), w^, cq ; 
Index 4, w”., I(^(;,, J aq), a., iv^, ; 
Index 5, J {w., ??/h), J (w., iv^), ; 
Index 6, 
Index 7, w'^, tq; 
Index 8, J {w., tv'^), t.. 
54. It will be seen from these forms that all the invariants retained are linear in 
all the quantities L, M. N, P, Q, R, S, a, A y, 8, e, a, b, c, k, I, m, n which occur in 
them. This ijroperty facilitates the expression of any other invariant in terms of the 
various members; thus 
LN — iVP _ (wo, w'o) — J3 {(v., tv' ,) 
ae — 4;g§ -h 3y~ _ tv^fj^ — 4J (w., tvj f),, + 3iv4h'^ 
- ’ 
and so for others. Moreover, in the invariants which contain a, h, c linearly, the effect 
is that the derivatives of <f> of the second order (being the highest that occurs) are 
contanied linearly ; and in those invariants which contain k, I, m, n linearly, the 
effect is that the derivatives of of the third order (being the highest order’that 
occurs) are contained linearly, as well as those of the second order. 
Moreover, the forms can be used to obtain the value of any given invariant ; 
all that IS necessary for this purpose is to obtain the expression of the invariant in 
terms of the members of the selected aggregate, and to substitute the values of the 
members that occur. Thus, consider the simultaneous invariant 
a, b, c 
U M, N ; 
E, F, G 
when expressed in terms of the members of the aggregate, it is equal to 
w, < J I - J [w.^, w'o) I {w^, tv%)} 
+ “3 ^^^"2 J - ut'2 J (it’s, w".)}, 
3 E 2 
