66 Royal Society. 



product of the tangents of two angles, the difference 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 I was at first disposed to think it must have been known and 

 published previously ; and had I not been assured by several eminent 

 mathematicians that it had not been previously published to their 

 knowledge, I should not have ventured to put it forth as new. 



Supplement to the preceding Paper, 



Professor Stokes, D.C.L., has pointed out to me an extension of the 

 preceding theorem, viz. that at every multiple 'point in a plane 



curve which fulfils the condition — % + —, -? = 0, the branches make 



dx dy z 



equal angles with each other ; so that, for example, if n branches cut 



each other at a multiple point, they make with each other 2n equal 



angles of -. 

 n 



The following appears to me to be the simplest demonstration of 



the extended theorem. At a point where n branches cut each other 



the following equation is fulfilled by all curves : 



Let 6 be the angle made by any branch with the axis of x ; then 



( 



costf — + sin0 — j (b — 0. 

 ax dyj 



But in a curve which fulfils the equation — x + __r =0, we have 



dx 2 dy 



d , d 



dy * dx ' 



whence it follows that in such a curve the equation of a multiple point 

 of n branches is 



j(cos0+ V — 1 .sin0)^- U = 0. 



Choose for the axis of x a tangent to one of the branches at the 

 multiple point. Then it is evident that the preceding equation is sa- 

 tisfied by the 2n values of d corresponding to the 2nih roots of unity, 

 that is to say, by 



= 0, * 2 -l t &c.,...i^LZl)l; 

 n n n 



therefore the n branches make with each other 2w equal angles of-. 



n 

 Q.E.D. 



