OF A SURFACE, AND TITEIR GEOMETRIC SIGNIFICANCE. 
377 
Consequently 
1J2 iw., us) + wJ^R (us)] - J {tv., IV,) J {tv., w'.) 
+ --K-{P{iv., w'.) + w.m {tv'.)] 
W.~tv. ' ' ' 
IV O 
o 
^~^{tv.tv,a^ — V~us^) -T ] us)J {tv., tv'.) 
iv/tv, ^ ~ ' ~ 
U'., 
H- /-r Uv.U'’.l {w., w'„) ~ , 
tr.hv. ' - ~ - ~ 
(/K . 
so that J- is expressed in terms of the members of the retained aggregate. Sul)- 
stituting the values of the invariants in the equation and dividing out by V®B after 
substitution, we find 
e/K _ I dH _ /o _ 2 . (/ A \ 2 d /I 
ds p' ds ~ ' p' ' dt ' o'] r' dn ' 
P 
4 c/P, 
pi 'Brads' 
a pro])erty Avhicli can be changed also into other forms, hy using the relation in 
§ 36, which expresses in terms of j- [\) and otlier magnitudes. Efiecting 
this change, and substituting for ^> c(^)’ values in terms of 
derivatives of p^, p., p, we can express the relation in the form 
d 
ds 
Px pt \p P-2/A 
I L _ 2\ I (/ /'I \1 _ 4 dB 
r V Pi p. p' p" d'ud p'j ] BP" ds 
and tlierefore 
1 dr 
d" ds 
1 + >i 
' ' ff ^ 1 
Pi P: P P dn p 
2 dB 
Br' ds ’ 
or, what is the same thing. 
4('C = ,i+ ‘ 1 ^dn 
ds \t ' Pi P 2 P p' 'P 
2 dB 
BP ds 
dB 
“d,^(p') + (^ p'jp'' PBd.A 
Proceeding to construct the otlier invariant that was suggested in § 39, we have 
y „,,,vU^ = (NP - 2MQ + Lll) (- F4,„, + G4.,„) 
+ (NQ - 2MR + LS) (E.^,„ - 
Let u denote the leading coefficient on the right-hand side, so that 
u = SEL - R (2EM 4- FL) + Q (EN -P 2FM) - PEN ; 
VOL. cor.—A. 
3 c 
