|
<, p' =
d[psi]/dx', q' = d[psi]/dy', which then will identically satisfy the
transformed equations F' = 0, G' = 0, H' = 0. The equation F' = 0, if
x', y', z' be regarded as fixed, states that the plane Z - z' = p'(X -
x') + q'(Y - y') is tangent to a certain cone whose vertex is (x', y',
z'), the consecutive point (x' + dx', y' + dy', z' + dz') of the
generator of contact being such that
/dF' /dF' / / dF' dF'\
dx'/ -- = dy'/ -- = dz'/ ( p'--- + q' --- ).
/ dp' / dq' / \ dp' dq'/
Passing in this direction on the surface z' = [psi](x', y') the
tangent plane of the surface at this consecutive point is (p' + dp',
q' + dq'), where, since F'(x', y', [psi], d[psi]/dx', d[psi]/dy') = 0
is identical, we have dx' (dF'/dx' + p'dF'/dz') + dp'dF'/dp' = 0. Thus
the equations, which we shall call the characteristic equations,
/dF' /dF' // dF' dF'\ // dF' dF'\
dx'/ --- = dy'/ --- = dz'/( p' --- + q'--- ) = dp'/( - --- - p'--- )
/ dp' / dq' / \ dp' dq'/ / \ dx' dz'/
// dF' dF'\
= dq'/( - --- - q'--- )
/ \ dy' dz'/
are satisfied along a connectivity of [oo]^1 elements consisting of a
curve on z' = [psi](x', y') and the tangent planes of the surface
along this curve. The equation F' = 0, when p', q' are fixed,
represents a curve in the plane Z - z' = p'(X - x') + q'(Y - y')
passing through (x', y', z'); if (x' + [delta]x', y' + [delta]y', z' +
[delta]z') be a consecutive point of this curve, we find at once
/dF' dF'\ /dF' dF'\
[delta]x'( --- + p'--- ) + [delta]y'( --- + q'--- ) = ]]>
|