Royal Society. 65 



places before it can be received as generally true. This seems 

 indeed but reasonable. At present, therefore, I will only ask 

 that my result may be looked upon as a protest against the oppo- 

 site and more generally received opinion. 



Greenwich, June 22, 1867. 



IX. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from vol. xxxiii. p. 554.] 



May 9, 1867. — Lieut. -General Sabine, President, in the Chair. 



THE following communication was read : — 

 J2 . 72 - 



"On a Property of Curves which fulfil the condition— -2 + —^ = 0." 



ax ay 



By W. J. Macquorn Rankine, C.E., LL.D., F.R.SS.L. & E. 



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

 Magazine 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 — ® + —2 =0 cuts itself in a double 



pointy it does so at right angles. 



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

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

 by the equation 



dx ay 



also that at a double point -2 and — t both vanish, so that the incli- 

 dx dy 



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



the quadratic equation 



dx~ dxdy d\f 



dy 



whence it follows that the product of the two values of -f-, which 



1 ax 



are the two values of the tangent of the inclination to the axis of x 



dx 1 

 !s = — — . In a curve which fulfils the before-mentioned condition, 

 a 



If 



the value of that product is — 1 ; and when such is the case with the 



Phil Mag. S. 4. Vol. 34. No. 227. July 1867. F 



