PROFESSOR A. R. FORSYTH OX THE DIFFERENTIAL INVARIANTS 
;!80 
I (/B 
and tlie expression can l)e further modified hj substituting the value of ^ given 
in § 40. 
42. As the quantity H, the measure of the mean curvature of the surface at the 
point, has occurred in the invariants in and a.i, and as the (luantities - and 
(Is an an as 
are not ecpial to one anotlier, we constiaict the quantities 
(/ /’a'\ 
dad A-’/’ d,s'\V^'' dnKV^I' 
It will a])pear tliat. hy means of the seconrl and the third of these, ve can obtain the 
(dH ^ d-H 
(Is (In (In (Is 
AVe liave 
'' 'L ) = d !(Er - 2F;8 + Ga) + ...} 
value of 
(Is ' A 
y 
+ F {3 ("A 
\ 
Let 
then as 
+ [a,J {iv,. n-A) - ina'A}. 
fq = (Ey-2F^ + Ga)<^or+^-- ; 
(E^ _ Fa)2 = K« (Ey - 2E;8 + Ga) - {ay - ^-) E' - (EG - F-) «n 
we 
have 
J- (an, tCj.) = n’., — iryH (aq) - Y~n' f. 
Heiice the invariant f), is expressible in terms of the members of the system; when 
the expression is substituted al)Ove, the result enaldes \is to obtain the value of 
H( a-’^) A" But it is simpler to modify tlie original system of concomitants in ^ 21 ; 
we can rejjlace EL (?aq) in the aggregate by hi, and the modified aggregate is still 
complete. 'J'he index of f)i is manifestly 4. 
When the various values are sul)stituted, we find 
G ^ _ iKiH _ d 
yr 
1 (h~ 
P 
I dHl 
p" cin y 
43. AA'A have 
,) := y3 [{E (G/3 - 2Ey + ES) - F (Ga - 2F^ + Ey)' 6,~ + . . .] 
I 
4- ^ [; aql (a'.,, ir".,) ■— A^hr'h' iq ~ J (au, aG,) ii,] 
H(aG) 
y 8 
J (/(v,, a’h)- 
