Mr Bateman, The determination of curves, etc. 163 



The determination of curves satisfying given conditions. By 

 H. Bateman, Trinity College. 



[Head 4 May 1903.] 



Since the co-ordinates of a point on a curve only involve one 

 parameter, it follows that a line associated with a point on the 

 curve will not take up every direction in space, but only a singly 

 infinite number of directions. 



Accordingly, if from a fixed point we draw lines in the par- 

 ticular directions, they will all lie on a certain cone which will 

 cut a unit sphere with the fixed point as centre in a curve. 



This curve has been called the indicatrix of the former curve ; 

 when the associated line is the tangent, principal normal, or 

 binomial at the point, the curve will be distinguished as the 

 tangent, principal normal, or binomial indicatrix. 



Corresponding to a given indicatrix we have a group of curves 

 which possess certain properties in common : for example, if the 

 tangent indicatrix is a small circle, all the curves will be helices. 



If we start with a given curve and derive a series of curves 

 from it by some definite process, the tangent indicatrix of the 

 original curve will in general be an indicatrix (for some associated 

 direction) of the derived curves. For example, if we consider 

 the evolutes of a curve, the principal normal of an evolute is 

 parallel to the tangent to the curve ; accordingly the tangent 

 indicatrix of the curve is the principal normal indicatrix of the 

 evolutes. 



One method by which a series of curves can be derived from 

 a given curve is as follows. 



Let be the origin, P a fixed point on a curve, Q a variable 

 point ; G the centre of gravity of a weight w at and the arc 

 PQ, where the density at distance s from P (measured along the 



curve) is -j- : w being any function of s. If the co-ordinates of P 



and Q are (a, (3, j) and (w, y, z), the co-ordinates of G will be 

 (£ V, where 



f *. [ Q du -, \ 

 v {w + u\=J y^ds Y (1), 



t\w + u}=\ z~j- ds 

 1 ' J P ds ) 



the constant in u having been chosen so that u vanishes at P. 



