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XI. On a new Form of Tangential Equation. By John Casey, LL.l )., F.B.S., 
Professor of Mathematics in the Catholic University of Ireland. 
Received January 24, — Read February 22, 1877. 
INTRODUCTION. 
Art. 1. The tangential equation of a curve is, as is well known, a relation among the 
coefficients in the equation of a variable line, which being fulfilled, the line must be a 
tangent to the curve. 
Let O be the origin, OX, OY the axes ; and let a variable 
line MN in any of its positions make an intercept v on 
OX and an angle <p with it; then the equation of the 
line is 
sc-\-y cot <p — v=0, 
and v and <p, the quantities which determine the position 
of the line, may be called its coordinates. From this 
it follows that any relation between v and <p, such as 
>=fW, 
will be the tangential equation of a curve which is the envelope of the line. 
This form of equation will be the special subject of this paper. Occasionally our 
investigations will embrace collateral subjects, when their importance will be such as to 
justify the digression. 
It will be seen that our form of equation admits of easy transformation into all the 
known forms of equation ; that it adapts itself with great facility to the various problems 
of the Integral Calculus relating to curves, such as Rectification, Curvature, Involutes, 
&c., and gives its results in very simple forms. 
In most of the methods of Modern Geometry, such as Pedals, Parallel Curves, Reci- 
procation, &c., it solves in a very simple manner problems that are very difficult by any 
other method. I have illustrated it throughout by numerous examples, most of which 
are of historical interest. Some of the problems discussed are, I believe, now solved for 
the first time, among which I may mention the rectification of Bicircular Quartics by 
Elliptic Functions. To this outline of the subject of this paper I may add that the 
MDCCCLXXVII. 3 G 
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