766] ON THE GEODESIC CURVATURE OF A CURVE ON A SURFACE. 329 
Addition.—Since the foregoing paper was written, I have succeeded in obtaining 
a like interpretation of the term 
V (p'q" -p'q) + jfp'q' (.Lp + Mq), 
which belongs to the general case. I find that these terms are, in fact, = — <£ + o) x p'; 
or, what is the same thing (since to = <£ + 6 and therefore co x p + to 2 g' = <j> + 6), are 
= 0 — to 2 q. It will be recollected that $ is the inclination of the curve to the curve 
q = c, which passes through a given point of the curve, <£> is the variation of </> 
corresponding to the passage to the consecutive point of the curve, viz., (f> + <j)ds is 
the inclination at this consecutive point to the curve q—c + dc, which passes through 
the consecutive point; to is the inclination to each other of the curves p = b, q = c, 
which pass through the given point of the curve, (o x the variation corresponding to 
the passage along the curve q = c, viz., co + co x ds is the inclination to each other of 
the curves p = b + db, q = c; and the like as regards 6 and to 2 . 
For the demonstration, we have, as above, 
]T p ' _ V 
<b = tan -1 „ , ~rr., to = tan 1 , 
Y Fp +Gq F 
where 
V=\/EG-F 2 ; 
and moreover Ep'* + 2Fpq + Gq - = 1. In virtue of this last equation, 
V>p'* + (Fp+Gqy=G; 
and we have 
—T(p'f-p”i0 + ^0, 
□ = (Fp' +Gq')p'V- Vp' (Fp' + Gq'); 
or, since V 2 = EG — F 2 , and thence 2VV=GE—2FF+EG, we have 
□ = i^ {(Fp' + Gq') (GE - 2FF+ EG) -2 (EG- F 2 ) (Fp' + Gq')}. 
Substituting herein for E, F, G their values E Y p + E 2 q', F x p + F 2 q', G x p' + G 2 q', the 
term in { 1 becomes 
= Ip' 2 + Jp'q' + Kq' 2 , 
where 
I = FGE X - 2 EGF X + EFG X , 
J = GE x - 2FGF X + (-EG + 2F 2 ) G x + FGE 2 - 2EGF 2 + EFG 2 , 
K= GrE. - 2FGF 2 + (-EG + 2F 2 ) G 2 . 
y 
But from the equation co = tan“ 1 p, differentiating in regard to p, we obtain 
“■ = mv (FGt: ~ 2EGf+EF6) =mvi ] 
C. XI. 
42