His 1892 textbook on applications of elliptic functions is of acknowledged excellence. He was one of the world's leading experts on applications of elliptic integrals in electromagnetic theory.[3]
He was a Plenary Speaker of the ICM in 1904 at Heidelberg[4] (where he also gave a section talk)[5] and an Invited Speaker of the ICM in 1908 at Rome, in 1920 at Strasbourg,[6] and in 1924 at Toronto.
In 1879 Greenhill calculated complicated twist rate formulas for rifled artillery by approximating the projectile as an elongated ellipsoid of rotation in incompressible fluid (which, as he couldn't have known back then, assumes subsonic flight).[7][8] Later, English ballistician F. W. Jones simplified it for typical bullet lengths into a rule of thumb for calculating the optimal twist rate for lead-core bullets.[9] This shortcut uses the bullet's length, needing no allowances for weight or nose shape.[10] The eponymous Greenhill formula, still used today, is:
where:
C = 150 (use 180 for muzzle velocities higher than 2,800 ft/s)
D = bullet's diameter in inches
L = bullet's length in inches
SG = bullet's specific gravity (10.9 for lead-core bullets, which cancels out the second half of the equation)
The original value of C was 150, which yields a twist rate in inches per turn, when given the diameter D and the length L of the bullet in inches. This works to velocities of about 840 m/s (2800 ft/s); above those velocities, a C of 180 should be used. For instance, with a velocity of 600 m/s (2000 ft/s), a diameter of 0.5 inches (13 mm) and a length of 1.5 inches (38 mm), the Greenhill formula would give a value of 25, which means 1 turn in 25 inches (640 mm).
Recently, Greenhill formula has been supplemented with Miller twist rule.
^Mosdell, Matthew. The Greenhill Formula. "Archived copy". Archived from the original on 18 July 2011. Retrieved 19 August 2009.((cite web)): CS1 maint: archived copy as title (link) (Accessed 2009 AUG 19)