TRANSACTIONS OF THE SECTIONS. 11 
a . © 
through the centre of gravity of A, where is given by the equation oat, It 
was also shown that Newton's construction affords, in certain cases, an easy method 
of estimating the directive effect of one magnet on another. 
On Statical and Kinematical Analogues. By Professor J. D. Evererr. 
If we take a line AB to represent a force along AB, the moment of this force 
round any point P will be represented by double the area of the triangle A BP. 
If we take the same line A B to represent a velocity of rotation round A B, the 
same double area will represent the velocity of P due to this rotation. 
We have thus a direct proof that a force acting along a line is the analogue of a 
velocity of rotation round it. Bf 
By supposing the line to be indefinitely distant, we obtain a couple as the analogue 
of a velocity of translation. 
The moment of a force round a line is the analogue of the component velocity, 
along this line, of any point upon it; and the resultant moment round a point due 
to any combination of forces and couples is the analogue of the resultant velocity of 
a point due to any combination of velocities of rotation and translation, 
In these statements, the moment of a force round a point is not regarded as a mere 
magnitude, but as a quantity having direction; in other words, as the moment of 
a couple whose plane.passes through the point and the line of action of the force, 
The velocity which is the analogue of the moment will coincide in direction with 
the axis of this couple. 
The only kinematical principles required for the demonstration of the above ana~ 
logies are (1) the parallelogram of velocities for a particle, and (2) the proposition 
that the velocity of a point due to rotation round an axis is perpendicular to the 
plane of the point and axis, and proportional jointly to the angular velocity and 
the distance of the point from the axis. 
On a New Application of Quaternions. By Professor J. D, Evererr. 
If w denote a velocity of translation, regarded as a vector, and o avelocity of rota- 
tion round the origin, represented by a vector drawn along its axis, we may-write. 
p=aotVap, oii... PEN OE NA 
where p denotes the velocity of the particle whose vector is p. Hence the symbol 
EMG Gab ei itieovnice tt e/a ae cilew dieleicceen (OP 
is a complete representation of the velocity of a rigid body. : 
Again, if » denote a couple (represented by its axis), and o a force at the origin, 
the value of p in equation (1) is the resultant moment round the point whose vector 
isp. Hence the above expression (2) represents a system of forces acting on a rigid 
dy. 
The expression (2) affords remarkable facility for the discussion of such subjects 
as the composition of velocities of a rigid body, the general properties of systems of 
forces, the conditions of equilibrium of a rigid body under constraint, and the rate 
at which a system of forces does work upon a moving body, | 
The author is developing the method in a series of aly in the ‘ Messenger of 
Mathematics,’ commencing with the Number for July 1874. 
On Partitions and Derivations. By J. W. L. Guatsner, M.A. 
It is well known that the number of partitions of x into the elements 1, 2, 3,..,. 
(the quotity of n with respect to 1, 2,3,.... i ie to Sylvester) is equal to the 
coefficient of x” in the expansion of (i=*) oe") (1-2) so ie the theory 
cae : 9! 
