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tangents to all the ellipses which are projections of the lines of cur- 
vature. For let a’ be eliminated by the equations 
ax? +4 2g? — @'2¢'” 
Sea” + a®h’c” = eatc? 
The result is 
a’b’e’4 + (fe°x’—a2h°z°—ea'e?) ¢’2+ eatc?z? — 0 
considering ¢” variable, this equation represents all the ellipses which 
are projections of lines of curvature. To determine the equation 
of the lines which enclose the space occupied by these ellipses let 
this equation be differentiated, considering ¢” as variable, the result 
of which is 
2a*b’c’? +. fc°x?—a°h?x’—ea‘e? = 0 
Eliminating ec” by this and the former, 
4ea*b°c*z’—(fc'a°—a°b?z°—ea‘c*)? = 0 
This can obviously be resolved into the factors 
2feabez + feea°—a'h’z’~ea'c? ~ 0 
2Je Phez—ferx? + a2b’s? 4ea'e? = 0 " 
And these again may be resolved into the simple factors 
Jf. cut abz— Je. atc =0 
Jf. cx—abz+,/e. ac —0 
Jf. ca +abs+./e. a2e =0 
—J/Ff. cx + abz+,/e. a’c =0 
a2 
