81 



the number expressing the number of the term so added and the 

 number next below it. 



If all the roots are impossible, the series will be either always 

 greater or always less than p n , whatever be the number of terms 

 taken . 



For an example take the cubic 



x z -\- px 2 -\- qx -\- r=0. 

 Here — r=3 x 1.2+ 2(p — 3)1 -f q—p+ 1 



+ 3x2.3 + 2(p-3)2 + q-p+l 

 + 3x3A + 2(p — 3)3 + q—p+l 

 4- &c. to x terms, 

 if x is a positive integer. 



The method in common with other experimental methods applies 

 to the discovery of all roots, possible or impossible, which do not 

 involve surds. 



March 12, 1849. 



On the Intrinsic Equation to a Curve, and its application. By 

 the Master of Trinity. 



The author remarked that the expressions for the lengths of 

 curves, their involutes and evolutes, in the ordinary methods, are 

 complex and untractable, which arises in a great measure from the 

 properties of extrinsic lines being introduced, namely, coordinates. 

 But a curve may be represented without any such additions, by an 

 equation between the length and the angle of flexure, which is 

 therefore called the intrinsic equation. This equation gives, with 

 remarkable facility, the radii of curvature ; involutes and evolutes 

 of most curves. It also expresses very simply what may be called 

 running curves ; namely, curves which run like a pattern along: a 

 strip of ornamented work. A very simple equation expresses, for 

 instance, the inclined scroll pattern so common in the antique, and 

 by altering the constants, gives to this pattern an endless variety of 

 forms. If s be the length of the curve and <p the angle, the intrinsic 

 equation to the circle is s = a<p; to the cycloid s= a sin (p. The 

 equation to an epicycloid or hypocycloid is s = « sin mp, according 

 as m is less or greater than unity. The equation to an undulating 

 pattern is <p = m sin s, which assumes very various shapes by varying 

 m. The method was also used in proving that if we take the suc- 

 cessive involutes of a curve an indefinite number of times, the re- 

 sulting curve (with certain limitations) tends to become the equi- 

 angular spiral if the unwrapping be always in the same direction, 

 and tends to become the cycloid if the unwrapping be alternately in 

 opposite directions. The latter proposition had already been dis- 

 covered by Bernouilli and proved by Euler. 



