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PROFESSOR. J. CASEY ON A NEW 
form of equation is suggested by Kinematics. For if we differentiate the equation 
vz =f(<p) with respect to a variable t (denoting the time) we get 
it=mt ( 2 ) 
Now if we suppose a rigid body to move so that a fixed point in it, say the centre of 
gravity, describes a right line, then ^ will be the linear velocity of the centre of gravity, 
and ^ will be the angular velocity with which the body revolves round the same point. 
Then the equation (2) will be the most general equation of the motion of such a body. 
It gives linear velocity divided by angular velocity as a function of the angle through 
which the body has rotated. From this it will be seen that some of our results will 
have a physical as well as a purely mathematical interest. With these remarks we pro- 
ceed to the subject of the paper. 
CHAPTER I. 
Section I. — Transformation of Cartesian into Tangential Equations. 
2. Definition. — We shall find it convenient to call the line OX, on which the 
variable line makes the intercept v, and with which it makes the angle <p, the director 
line. 
3. If the Cartesian equation of a curve be TJ=0, we can by the usual process find the 
condition that the line x-\-y cot <p — v =0 touches it; this condition will be our tangential 
equation. For this purpose the equation of the line may be written in the form 
( 3 ) 
where t denotes tan <p ; and eliminating y between this and the equation U = 0, we shall 
have an equation in x of the form 
(AcA, A 2 ,...A„X^— l) n =0 (4) 
The discriminant of this will be the tangential equation required. It can be transformed 
into the usual form of tangential equation by changing v into — v ~ and t into \ This 
is evident by comparing the equations 
x-\-y cot <p — !/=0, \x-\-g>y-\~v= 0. 
Cor. The usual form of tangential equation can be transformed into our form as 
follows : — Let 
4/(x, [A, y) = 0 (5) 
be the tangential equation, say cf the ?ith degree; divide by X”, and change * into —v, 
and ^ into cot <p. 
