OF A SUEFACE, AND THEIE GEOMETEIC SIGNIFICANCE. 
387 
Invariants involving Derivatives of Two Functions. 
49. Among the aggregate of invariants set out in § 23, there still remain nine 
as yet iminterpretecl; l)ut their expressions involve derivatives of the function i//. 
Five out of these nine involve derivatives of no higher order than those which occur 
in the invariants interpreted in 34—38. In order to obtain their inter]fretatioii, it is 
convenient to associate with them invariants which depend upon i// alone and bear 
the same relation to alone as some of those already interpreted bear to <f) alone ; 
and then the complete aggregate can be simplified by replacing some of the original 
forms by some of the associated forms. 
For this purpose, let 
''uD 
Wo = (E, F, GJi/zoi, — 
J (Wo, W h) = (E/> — Fa') — (EA — Ga') i//oiiAio + (Fc' — Gb') xp^Q~, 
= - (Eb — F(d) (poi^oi — (Eg' — Grt') (<'/)oi’Aio + 
+ 2 (Fg' — Gb') (pirfj,); 
then we establish (and it is easy to verify) the equations 
J~ (u\, 7Co) — aqW.o + Vhcp = 0, 
^CoAi 7V oJ (u'j, "ICo) -|- (iV.T, 7c'"o) — 0, 
- w'",W'", + ufH (?c'",) = 0, 
~ "J j I (^G’ ^F"o) = 2V~7epc'"o, 
A^' - 4J (aq,iF"o) J (Wo, W'G) = {P(wo, w'G) - 4V~H(?o'".,)}. 
The first of these equations shows that Wo can be regarded as known ; it is not an 
independent invariant hut, if we wished, we could replace 7 Co by W^o in the complete 
aggregate without affecting the completeness. This change will not be made; we 
shall letain Wg as a quantity convenient for other purposes and alternative to an in 
the aggregate. 
The second and the third equations, combined so as to eliminate A, show that W'"o can 
be regarded as known ; it is not an independent invai iant but, if we wished, we could 
replace w by W' h in the complete aggregate witliout affecting the completeness. 
This change will be made. 
The fourth and the fifth equations, combined so as to eliminate A.,, show that 
J(ffo, A''"o) can be regarded as known; it is not an independent invariant but, 
if we wished, we could replace J (w.^, w'"^ by J (Wo, W'"2) in the complete aggregate 
without affecting the completeness. This change adso will be made. 
3 D 2 
