OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
391 
bi = (GP — 2FQ + ER) ijjQi + (GQ — 2FR ES) (— ^iq), 
I)o = (GP — 2FQ -|- ER) (— Fi/zq]^ + Gi/z^u) 
+ (GQ ~ 2FPt + ES) (Ei//u^ — ! 
then 
bi _ (/H R — A —^ 
Zc' ’ V' “ (h? ' 
Now we have the equations 
= J (u’l, iv.^ — it'iao 
Hqlq = + J {n\, W.,) in 
which are easily established; substituting in them the values of tlie Invariants that 
occur, we find (on removing a factor AB“V^), the relations 
(/H ^ r/H 
d/ “ <h 
(/H ^ <m 
(In' (Is 
cos X ~ 
sin X + 
r/H 
(In 
sill X 
c/H 
(In 
cos X 
t. 
J 
which are the ordinary differential relations for transference from directions'^' ds and 
dn to ds' and dn', when the subject of operation is a function of position only and 
involves no properties of tangency and no properties of contact of order higher than 
the first. But for a function of position (and, a fortiori, for a function w^liich involves 
properties of contact of the first order or of higher orders), the operators and 
are not interchangeable. Thus, in particular, ^ 7 ^ dndfs equal to one 
another, except for special curves ; an expression for their difierence has already been 
obtained. 
52. It still remains to identify the four invariants H {w"f), J (('q, 
which involve the derivatives of both (j) and if/. Instead of proceeding to obtain 
their values, we use the method adopted in § 49 ; we replace them by four equivalent 
invariants involving derivatives of ijj only, and the change does not affect the 
completeness of the aggregate. These four invariants are 
W"3 = (P, m', n'XAn, - ^/bo)^ 
II (W-Q = (k'm' - n if/of + ... 
CR (W'f) = {k'^~n’ - mfrn' + 2P) if/^f + . . ., 
J (W., W"3) = {YJ - FP) if/.f + ... 
* The value of sin A shows that the direction of dn falls within the angle between ds and dn. 
