1 58 Royal Irish Academy. 



fluid of greater refractive power than itself. It is to be applied suc- 

 cessively to the ordinary and extraordinary refracted vibrations, and 

 we thus get the uniradial incident and reflected vibrations, or rather 

 the ellipses which are similar to them. And as any incident vibra- 

 tion may be resolved into two which shall be similar to the uniradial 

 ones, we can find the reflected vibration which corresponds to it, by 

 compounding the uniradial reflected vibrations. 



It may be well to mention that, in a uniaxal crystal, the plane of 

 the extraordinary refracted vibration is always perpendicular to the 

 axis, and therefore the ellipse in which the vibration is performed 

 may be easily determined by the principles already laid down. The 

 plane of the ordinary vibration has no fixed position in the crystal ; 

 but if we conceive the auxiliary quantities £,, «„ £, (Phil. Mag. S. 3. 

 vol. xxi. p. 230) to be compounded into an ellipse (as if they were 

 displacements), the plane of this auxiliary ellipse will be perpendi- 

 cular to the axis of the crystal. 



Whether the preceding very simple construction, for finding the 

 incident and reflected vibrations by means of the refracted vibration, 

 extends also to the case of biaxal crystals, is a point which has not 

 yet been determined, on account of the complicated operations to 

 which the investigation leads, at least when attempted in any way 

 that obviously suggests itself. 



A paper was read by William Roberts, Esq., F.T.C.D., " On the 

 Rectification of Lemniscates and other Curves." 



Let a curve be traced out by the feet of perpendiculars dropped 

 from a fixed origin upon the tangents to a given curve : and from 

 this new curve, let another be derived by a similar construction, and 

 so on. Also let a curve be imagined which is constantly touched by 

 perpendiculars to the radii vectores of the given curve, drawn at the 

 points where it is met by these radii, and from this let another be 

 derived by a similar mode of generation, and so on. 



Then if s n denote the arc of the curve which is rath in order in 

 the former series, and s_„ that of the rath in the latter, we shall have 



d : w /1 , . dw . . dou 3 

 _ ±""& + <- 1 ± n >l? + r '3* / du,\±-'rdr. 



F (r, w) = being the polar equation of the given curve. 



It is convenient to distinguish the curves of the two series by 

 calling those of the former positive, and those of the latter negative ; 

 we may also generally denote their polar coordinates by the symbols 



If the given curve, which may be denominated the base of either 

 system, be an ellipse whose centre is the origin, it will be found, by 

 applying the above formula, that the negative curves will in general 

 have their arcs expressible by elliptic integrals of the first and second 

 kinds, whose modulus is the eccentricity of the base-ellipse. The 

 arc of the first will involve only a function of the first kind : a result 



