OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
383 
The actual values are found to lie 
(* S 
- i H (K„4,„f - 2F„A,.'#-,o + Gv.<i>vr) 
+ 
i ■/'"' A’ (E„,</.„!= - ■2Y„<I>,,<I>„ + GM), 
^ ^ .1 
—3( — <f>oi^-2() + — </>io¥(ie) 
(IS- iC.A 
+ i feU' (E,rf„r - 2F,„c^„,/,,„ -L 
(E,„<)!>,/ - + (i„y<t,yr)- 
?r,v 
where r and s on the ri^ht-hand aides liave the values o-iven in 5 15. Now 
o o • 
V= ''T = + 2Taii,T'^' + var,,,;/'- + V=H„,F' + Ym„,y" ■ 
when we substitute the values of the various (juantities and reduce, we have 
'' ^ p" (In 
_ 1 l)i ^ r K I Id 
(h~ B- ^ ^ 
P 
the same value as before (§ 42). 
Again, we know"^ that 
(lx 
(It 
fly / 
a, ="' 
df = tj" + 2r'.T'i/ + 
(It-' 
(It 
■i = Ax'- + 2A'xb/' + 
Hence, as 
rim 
(If 
rPx 
= iL„r= + 2H, ,*y + + h,„ + h, 
H)1 
rPy 
we 6nd, after substitution. 
Consequently wx have 
r/'H _ I III -L 1K (H 
(it 
rUH rim 1 r/H 
p" (hi 
;/y hih 
* See the paper by the author, quoted in § 4. 
