326 
PKOFIiSSOIi A. R. FORSYTH ON THE 
where fl" is tlie angle at which the surfaces intersect at the point, that is, the angle 
between the normals to the surfaces. Thus 
1 ) 0 , 
IT 
(1(f) (1(f)' 
(In dn' 
cos o' 
Furtlier, let fi be the angle at which the two surfaces (f)' = constant, (f)" = constant, 
intersect; and let Dd Ije the angle at which the two surfaces (^ = constant, 
(f," = constant intersect. Then we have similarly 
_ d(f) d(f)' 
dn dn" 
cos nf 
1)1 )T), ^ d(^ d4" D'-@, _ (d(f)"V 
L-^ dn' dn"^^"-' ~ \dJ(")' 
31. Again, let ds denote an element of arc through the point along the curve of 
inteisection ol the suriaces (f) = constant, (f>" = constant; let dd l)e a similar element 
along the intersection of <^ = constant, (jf)" = constant; and let <h" be a similar 
element along the intersection of (f) = constant, (f)' = constant. Then 
// du . I, do . 
dtv 
ds 
0 , 
and therefore 
say, Avhere 
N (JAV 
so that 
J_ A 
ch + ^ 
n do 
d.s- 
+ 4>"(m 
II 
1 du 
1 do 
L div 
0^ ds 
do ds " 
dg ds 
T, 0o = 
' ’j 
4^ lUD ’ 
^ 1)10 V 
4* 001 
9 luo’ 
4>'\.n 
('6 I>, (\ J\ p, hjj.lu, di\ dw)- = ds', 
<■>/ if, ^ 3 )" 
say. 
Again, we have 
= (V.u 
1_ du 1 dv 1 d(V 1 
d, (Is $2 ds 0.^ ds 
?t>3 
= (a, h, c,f, y, h'le,, 0. , 0^Y 
-J- •^3^2 + ^3^3)“ + (i/1^1 + yY^-i “f .Vs^s)" + (^ 1^1 + 
+ ^ 3 ^ 3 )'- 
