Hamilton's Hat Trick

  Hamilton performed a bit of a hat trick when deriving the conditions for the integrability of a first order differential. In subsection [8] he makes an observation about the unit vector for the direction of a ray and then states three conditions that must be satisfied. Where do they come from? We can start with deriving expressions for the direction cosines α, β, & γ and see what happens to the magnitude of the unit vector as we vary the point of intersection.

The result is a condition involving the direction cosines and their gradients but it isn't quite what we're looking for. We can next try evaluating the gradient of one of the direction cosines and see what happens.

We find that the gradient can be expressed as the sum of two vectors. The same can be done for the other gradients and taking the dot product of each of the gradients with the direction unit vector we get Hamiton's three equations.

Supplemental (Jun 9): We can verify the first condition found by using the formulas for the direction cosine gradients.

Hamilton appears to have arrived at his set of equations by noting that the direction of change of the direction cosines is perpendicular to the direction of the ray so their projections onto the direction of the ray would be zero.