468 Prof. W. J. M. Uankine on a Property of Curves, [May 9, 



latioiis ; and I have not at present either the requisite leisure or materials 

 for following out so delicate an investigation. I trust that the facts already 

 elicited are sufficiently novel and important to justify my bringing them, as 

 they now stand, before the Society. 



II. On a Property of Curves which fulfil the condition 

 £1^ + ^ =0." By W. J. Macquorn Rankine, C.E., LL.D., 

 P.R.SS.L. & E. Received April 9, 1867. 



1. In a paper *' On Stream-Lines," published in the Philosophical Ma- 

 gazine for October 1864, I stated, and, in a Supplement to the same paper, 

 published in the Philosophical Magazine for January 1865, I proved the 

 proposition that " all waves in which molecular rotation is null begin to 

 break when the two slopes of the crest meet at right angles." 



2. I have now to state the purely geometrical proposition of which that 

 mechanical proposition is a consequence. If a plane curve which fulfils 



the condition ^-f + 4-^=0 cuts itself in a double point, it does so at 

 dx cfy 



right angles. 



3. The following is the demonstration. It is well known that the incli- 

 nation of any plane curve to the axes at an ordinary point is given by the 

 equation 



^dx-\-^dy = 0', 

 dx dy 



also that at a double point ^ and ^ both vanish, so that the inclinations 

 dx dy 



of the two branches to the axes are given by the two roots of the quadratic 

 equation 



dx dx dy dy 



whence it follows that the product of the two values of which are the 



dx 



d'(l> 



two values of the tangent of the inclination to the axis of^,is=^. In 



d 



dp 



a curve which fulfils the before-mentioned condition, the value of that pro- 

 duct is — 1 ; and when such is the case with the product of the tangents of 

 two angles, the diff"erence of those angles is a right angle ; therefore the 

 two branches cut each other at right angles. Q.E.D. 



4. The proposition just demonstrated is so simple and so obvious, that 



pials, corresponds homologically with that which, as a general rule, is most persistent 

 in the typical diphyodonts, including Man, viz. the posterior milk-molar, replaced by 

 the posterior permanent premolar. 



