1 'Tl'TERENTIAL TXVAKTANTS OF SPACE. 
Tlie former of tliese can l)e replaced dy the three ecpiations 
a? -{~ d“ •“ pU “h cl 
h/ + hm + fy; = pY d- o-m 
gl + fra + cn = pW + cr/i 
and wo also have 
/- 4- rrr + rr = 1. 
Let (Is ho an element of arc of a lino of curvature ; then 
(J [d4>\ — I j ^ \ ^ t 
*UJ “V cV- + ’'' -' + 
0 \ Cf/» 
?y 
zj dll 
d + 4 / + ^,3)^ 
_ 0 _i_ 0 , 0 
\ CX Cl/ 03, 
_ I (Ua 4 ^ h 4 “h (Uh 4 ^ h 4 W^f) 4 (Ue 4 4 
“ 2 (U~ 4 4 W3)i 
Multiplvino- tlie foregoing three erpiations ])y U, V, W and adding, we have 
d 
that is. 
p (U^ 4 V- 4 W-) = 2 (U- 4 V- 4 W2)-5 
„ d<ii _ d ld<f>\ _ 
' dn 
ds ' dn .'' 
\ _ .)g 
dYdrJ 
say. Multiplying the same three equations hy /, vyy, a and adding, we haam 
c — aL 4 21i/yyi 4 2g/?i. 4 byy^'' 4 2fyyyyy 4 cy/^ 
_ 2 0 (^ 
1V 0yy 
where II is the radius of curvature of tlie princijyal normal sectioii at tlie point. 
Thus p and c can l)e regarded as known. 
Again, solviypg the three equations for /, m, n, we olkain three results of the form 
a — O', 
h , 
E 
l = p 
u, 
h , 
E 
h , 
b — cr. 
f 
Ay 
b — cr. 
f 
E - 
f , 
C — cr 
W, 
f , 
C — cr 
AVhen these are substituted iii /- 4 ^id 4 = 1, we have 
a — 0-, h , g 
h , b — 0-, f 
E ’ f , C — (T 
* Frost, ‘ Solid Geometry,’ p. 290 . 
