564 



On a new Form of Tangential Equation, [Feb. 22^ 



body, it approaches S^^iocJiaiojjterus. While it conforms to the Spionidcie 

 in the structure of its body-wall and bristles, it diifers in regard to the 

 absence of the dorsal branchia? : and further, the short, pinnate and 

 ciliated anterior branchial organs of Priouosjoio appear to be the nearest 

 approach to its elongated tentacles. In the mechanism of its proboscis 

 and in the structure of its snout and circulatory organs, again, it presents 

 features sui f/enerh. 



III. On a iievr Form of Tangential Equation.-'^ By John Casey^ 

 LL.D.^ F.R.S.^ Professor of Higlier ]Matliematics in the 

 Catholic University of Ireland. Eeceived January 21^ 187r. 

 (^Abstract. ^ 



If a variable line make an intercept 7> on the axis of :o, and an angle (p 

 vith it on the negative side, the equation of this line ^^ill be 



^v-\-u cot (p — r = 0. 



The quantities y and (p will determine the position of the line, and may 

 therefore be called its coordinates : hence any relation between i' and 0, 

 such as r =/(&), will be the tangential equation of a curve which is the 

 envelope of the line. 



The equation v=f{<p) forms the subject of this paper. It is remark- 

 able for the facility with which it can be transformed into the ordinary 

 Cartesian and tangential equation, as well as into the polar and intrinsic 

 equation of a cmwe. In a great variety of cases it gives, in a simple 

 form, results which, by other methods, are very ciunbersome or nearly 

 ijnpracticable. I have illustrated it throughout by numerous examples, 

 many of which are of historical interest. 



The following is an outline of the contents of the paper : — - 



Chapter I. shows how to transform Cartesian and polar equations into 

 the form r=:t\(p). In the coiu'seof the investigation a remarkable system 

 of curves of the ufh class, which are concomitants to any curve of the 

 nth degree, are introduced, and their leading properties investigated. 



Chapters 11. . III. are occupied with the transformation of the intrinsic 

 equation, and vice versa, and some allied subjects. In these chapters the 

 whole theory of evokites, involutes, cmwature, &c. are fully considered. 



Chapters IV., T. are devoted to the investigation of the properties of 

 cycloids and hypocycloids by their tangential equations. A large number 

 of new properties of these curves are given. The following may be taken as 

 specimens : — 1st. If three tangents to a cycloid be given, the envelope of 

 the tangent at its vertex is a parabola. 2nd. If two tangents to a cycloid 

 contain a gi^ en angle, the locus of the centre of the circle described about 

 the triangle formed by the two tangents and their chord of contact is a 

 right Hue. 



Chapter YI. contains the theory of positive and negative pedals. The 



