This is called Abel's integral equation and allows us to compute the total time required for a particle to fall along a given curve (for which d ℓ ℓ / d y would be easy to calculate). But Abel's mechanical problem requires the converse – given T ( y 0 ) , we wish to find f ( y ) = d ℓ ℓ / d y , from which an equation for the curve would follow in a straightforward manner. To proceed, we note that the integral on the right is the convolution of d ℓ ℓ / d y with 1 / y and thus take the Laplace transform of both sides with respect to variable y :