OF A SURFACE, AND THEIR GEOxMETRIC SIGNIFICANCE. 
335 
Increments of the Arguments. 
7. We now require the increments of the various arguments, corresponding to tlie 
increments of x and y. We denote l)y E', F', . . . the same functions of X and Y as 
E, F, . . . are of x and y ; thus, if c/E be the increment of E, we have 
E' = E + dE ; 
and so for the other magnitudes. 
Since the relation 
E dx^ + 2F dx dy + G d^f = E' dX:^ + 2F' dX dY + G' dY'^ 
holds for all values of dx and dy, we have 
E = E' f + 2F' + G' 
■hx / ax ox 
= E' (1 + 2^jq dt) + dt, 
p ^ p,, 0X ax p,, /ax aY ax aY\ aY aY 
ax dy vax dy dy dx j dx dij 
= E'^oi + F' (1 + dt + 17131^ dt) + G't^iq dt. 
G = E' f + 2F'^-^ — + G' 
= 2F'^q3 dt + G'(l + 2rjQ^ dt). 
We thus have 
- dX = ( 2 E'^ 3 o + 2 F'. 73 o) 
Now, the differences between E and E', F and F', are small quantities of tlie order 
dt ; hence, when we are retaining only small quantities of the order dt on the riglit 
hand side, we can replace E', F', G' Ijy E, F, G respectively; and we find 
~ gf — 2E^1I3 + 2F773,j 
~ = F^o 3 + F^ju + - • 
- + 2G,„, 
J 
Similarly, the relation 
L dx^ + 2M dx dy + N dif = - = L' dX^ + 2M' dX dY + N' dY'^ 
