305 



7. Instead of eliminating R between the two equations (m), we may 

 legin by eliminating v ; a process which gives the following quadratic 

 in Rr x (the curvature) : — 



(q) . . (ell- 1 - efr) (e"R-> - e" = (J Br* - eK~ l Y ; 

 or (r) . . R 2 - FR X + G ; where (because ee" - e n - K 2 ), 

 (s) . . F = Ef l + Ri l = (eE" - 2e f E' + e"E) iT 3 , and 

 (t) . . G = R^R-i 1 = (_£#" - E n )Er*. 



We ought, therefore, as a First General Verification, to find that this 

 last expression, which may be also thus written, 



agrees with that reprinted in page 521 of Liouville's Monge, for what 

 Gauss calls the Measure of Curvature (k) of a Surface ; namely, 



DP" - JOfW 

 ^ * ' (AA + BJB+CC)* 1 



which accordingly it evidently does, because our symbols L M JSfA B C 

 represent the combinations which he denotes by ABCD D'D". 



8. As a Second General Verification, we may observe that if /be the 

 inclination of any linear clement, du = vdt, to the element du = 0, at the 

 point P, then 



Kv 



(w) . . tan I r ; 



v ' e + e v 



and therefore, that if H be the angle at which the second crosses the 

 first, of any two lines represented jointly by such an equation as 

 (x) . . Av 2 - Bv + (7=0, with v x and v 2 for roots, then 



(j) . . iun 77= to (/,-/,) .-i^-,/ - ; 



so that the Condition of Rectangular ity (cosJ2"=0), for any two such 

 lines, may be thus written : 



(z) . . eA + e'B j-e" C = 0. 



But this condition (z) had already occurred in No. 5, as an equation Q?) 

 which is satisfied generally by the Lines of Curvature ; we see therefore 

 anew, by this analysis, that those lines on any surface are in general 

 (as is indeed well known) orthogonal to each other. 



9. Finally, as a Third General Verification, we may assume x andy 

 themselves (instead of t and u), as the two independent variables of the 

 problem, and then, if we use Monge 's Notation ofp, q, r, s, t, we shall 

 easily recover all his leading results respecting Curvatures of Surfaces, 

 but by^transformations on which we cannot here delay. 



