DIFFERENTIAL INVARIANTS OF SPACE. 
321 ) 
4‘jry, 
4>.rz, 
4>.c 
) 
4^ijy> 
4>zy. 
4*y 
4^.vz; 
4^y-i 
4>zz. 
4>z 
4*y ’ 
4^z , 
0 
= — 4>r^) + • • • 
4*j-^> 
4>xy, 
4^.-z 
4*x^, 
4*yy> 
4>y= 
4>.rz, 
4>yz, 
<4>zz 
As usual, let lij and be the principal mdii of curvature of the surface 
(j) = constant; and let 
O' == p + 4^/ + ^Av)h 
l\j 
for the radii R, = K, and Rj = R. Tlieii'^' the two values of cr are given hy the 
equation 
and therefore 
b 
1 
4^^y 
4>.<-z 
4>.r 
= 0; 
4>xy , 
4^y>/ ~~ n", 
4^y~ ’ 
4>y 
4>z:z 
4*yz 
b 
1 
4>-. 
4^j- ? 
4*y ’ 
4>z 
0 
CTiCr. 
_ { 4 ^y!, 4 >-- ~ 4 >y=~) + • • 
+ <!>/ + 
Consequently 
' + 4>/ + <^v 
and therefore 
02 
4@iAi 
9, = = 
4>.-J^ + 4^f + </>p 
P\P-2 
02 _ , 0 
/\ 3 
_ 0^ I 
A I ll jR.i 
- I 
Ai^ R1R2 
4 / d(f) 
RjRo \dn/ ' 
* Frost, ‘ Solid Geometry,’ p. 293 
2 IT 
VOL. CCII.—A. 
