OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 401 
Also, we write 
= 4V% - 4V46' {2A' + 3r) - p, 
X, 4V% - 4V^6»(2r' + 3A") - q. 
Then a first expression for the differential invariant of deformation of the third 
order is found to be 
(EX/ - 2Fkf, + GXf) 
This expression can be modified by means of the relation (§35) 
4V^(LN - M") = V 
= — 2 V'^ + E [(Eq! — 2Fio) Goi + Gio^} +G [Eq/ —Eio(2Ffji — G^q)} 
+ F [EioGoi — Eqi (2Foi + Gio) + 2Fio (2Fy^ — Grio)}. 
Dividing both sides by and taking the derivative with regard to x, we find (on 
\ising the relations in § 6, and after reduction) that we have 
Xi = - 8V-^(NP - 2MQ + LR), = - 8V%, 
say. Proceeding similarly from the derivative with regard to y, we have 
X, = - 8V" (NQ - 2MR + LS), = - 8V*b, 
say. It thus appears that the two combinations E, F, G and their derivatives up 
to the third order, represented by X^ and X.,, are expressible in terms of the fundamental 
magnitudes of the second and the third order. Moreover, dropping the numerical 
factor 64, we have an expression for the differential invariant of deformation of the 
third order (say I) in the form 
IV® = Eb' — 2Fab + Ga~. 
By the theory in the preceding memoir, this invariant (which now involves only 
fundamental magnitudes of the first three orders and none of their derivatives) ought 
to he expressible in terms of the members of the system set out in § 53. Writing 
iv\ = (a, bX(/)oi, - 
w'\ = (Eb - Fa, Fh - GaX^on “ 4>io)^ 
we find 
wfw\ = (^ 2 , w'f) + ^^; 2 ^^;' 2 ^l — 2J {iv^, J '^%) — 2Yho\yo^, 
iv^w'\ = wff {w^, wf) J (iv^, wf) + 2V^'tCoJ (w^, w'f) — 2N‘^uf(iv^, wf) 
— 2 - 1 ^ 2 ^] J {'^ 2 ) ^^^ 3 ) + w<pv\a.i. 
3 F 
VOL. CCI.-A. 
