OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 373 
with the earlier significance for A, D, W, and T is given by 
T = (P, Q, R, Six', yj ; 
and another form is 
(Js X w.^/ V~a’A 
37. We require derivatives of some of the binary quadratics with respect to an 
arc in the surface normal to the curve = 0 ; for this purpose, we proceed as in § 36. 
We take 
and we have 
=f4>in — 2r7<^oi^io + H 
10 ’ 
cht 
— 2y</»on^ii (~ F^oi + •••) + ^03 (E(?5>oi + •••)} 
“ { — F^oi + •••) + <^11 (^<^01 + •••)} 
+ 4*Ql^{fl0 ("~ + • •) +/oi (E </>01 + ••)]+•' 
and so, after some transformation and reduction, we find 
= '/'oi' -1"'') - »(E*- - F“)i + • • • 
+ <!.„,= {,/'E (EG„, + FG,„ - 2FF„) - (/F + ^E) (EG,„ - FE,„) 
+ (— + 2EF10 — FEio) + (— Fy^o + E/oi)} +. 
Firstly, let f, (j, h = E, F, G, so that u becomes aq. The coefficient of the first term 
in the earlier aggregate is 
= EA - 2EF6 + F\^ 
= E (Ec — 2F6 + Gu) — Vkq 
and therefore that aggregate is 
= aql (aq, a/b) - Vha'b. 
The coefficient of the first term in the later aggregate is 
E - F (EGio - 2FFio + GE^o) + E (EGoi - 2FFoi + GEoO 
and therefore that aggregate is 
Consequently 
VViOj fb = ^ <'"2’ '"'y “ In ' 
(vS) = »"=) - v^«v. 
and therefore 
