8 
Proceedings of Royal Society of Edinburgh. 
SESS. 
der Waals’ equation it is shown that the ratio of the pressures 
required to reduce the volume of a fluid by 10 per cent, and 
by 5 per cent, respectively, cannot lie between about 2*3 and 2 ’8. 
But Amagat’s experiments show that, in all the seven liquids he 
examined (bodies as different as water, alcohol, and ether being 
included), the ratio in question varies from 2*5 to 2*73. 
In another paper, Professor Tait deals with the effects of the 
Rotation of a Projectile on the Form of its Path. He shows that 
the main facts were known to Newton, and that he gave the 
correct explanation ; that Robins, seventy years later, independently 
re-discovered, and gave beautiful experimental proofs of them, 
though, nowadays, Magnus alone is usually mentioned in con- 
nection with the subject. Professor Tait proceeds to show that 
the peculiar feather-like flight of a well-driven golf-ball, the 
initial upward concavity of its path, the length of time it remains 
in the air, and the consequent large increase of its range over that 
calculated from the ordinary theory of resisted projectiles, are due 
entirely to the rapid under-spin which the ball receives when 
properly struck. He also points out that, by sufficiently increasing 
the rate of rotation of a golf-ball, it may be made to move in a 
path which has a kink ! 
Dr Sprague’s paper, entitled a New Algebra, treats of some of 
the different kinds of operations which may be performed on 
permutations of the first natural numbers. It consists mainly of a 
development of the laws according to which the symbols denoting 
the seven operations combine with each other. It is found that 
these laws differ in several respects from the rules of ordinary 
Algebra, and any other calculus known to the Author, who there- 
fore thinks that the somewhat ambitious title of the paper — a New 
Algebra — may be justified. It appears, from the examples given 
of the application of his method, that it is very closely connected 
with the theory of numbers ; and one of the results obtained 
throws some light on the theorems of Fermat and Wilson. 
Dr Knott’s paper on Recent Innovations in Vector Theory is 
controversial in character. Its aim is to refute the arguments of 
certain critics who have asserted that some of the principles of 
Hamilton’s Quaternions are unnatural, paradoxical, or lacking the 
characteristics of fundamental geometrical principles. The Author 
