A Justification of Hamilton's Hat Trick

 Hamilton noted that the direction cosines do not change for variations along a ray. But getting from here to his three equations is a bit of a leap. One can show that the variations in the direction cosines, δα_i, are equal to a dot product and therefore proportional to the projection of the variation in position, δr̅ onto directed line segments A̅_i.

This shouldn't depend on what coordinate system we use so we can try to see what happens in spherical coordinates. We don't need to know the exact values of the components of the gradients A̅_i so we can represent them by a set of functions g. Since the direction cosines don't change for variations along the ray we conclude that g_ir=0 and the gradients are perpendicular to the direction of the ray.

Hamilton's methods appear to be essentially geometrical which employs projections and so he would not need a complete understanding of vector analysis to make some of the deductions.