A Catalogue of a Span Classsearchtermspan Classsearchtermcollections
A Catalogue of a Span Classsearchtermspan Classsearchtermcollections
Monsieur Fabre De Lagrange
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Differentiating implicitly a + l% = 0. (4. ) Proceeding to a second differentiation, in which, by virtne dy of the last equation, we may consider ~ as constant, we obtain dx' ^ dxdy \dx) ^ dif \dx) ^ ^ Then, if we take (2) as the director plane, that i^ to say, if we make the director plane parallel to the axis of z, we must substitute in (5) the value of -/ obtained from (4), and we dx get j^dh ci J d^z, 9 d^z _ b^-7-r, - 2ab -j-j- + «2-^, = 0. Dx-" dxdi/ ay- or in the usual notation Z>V - 2ah...s + a^t^ 0. (6. ) If, however, we take (1) as the director plnne, we mu^t determine j ^''^"^ ('^)) which gives dy _ _ C/? + A dx ~ Cq + B and the differential equation becomes (C^ + B)V - 2 (C7 + B) (Cp -[- A) s + (C;^ -? A)^ t = 0. (7. ) which reduces to the same form as (6) if C ■= 0. 30 whence and • Ruled Surface with right line Director, The axis of z being taken as the director, the equations of the variable line may be written as X ^^ ky mx -\- n = z. dy \ y j- = ~y z=^ = constant, (tX ft X dz dz dy dx dy dx Differentiating again, = ^+2 — -^ + ^(''^V dx'^ dxdy dx dy'^ \dx) 1 % u dy Substituting - for -^ we obtain ^ x ax „ d^z, ^ d^z „ d^z ^, ^ which is the equation of the surface.
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