1867.] Prof. W. J. M. Rankine on a Property of Curves. 469 



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. Received April 23, 1867. 



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

 ceding theorem, viz. that at every multiple point in a plane curve which 



fulfils the condition — %■ + —2 =0, the branches make equal angles with 

 dx^ dy^ 



each other ; so that, for example, if n branches cut each other at a multiple 



point, they make with each other 2« equal angles of -. 



n 



The following appears to me to be the simplest demonstration of the ex- 

 tended theorem. At a point where n branches cut each other the follow- 

 ing equation is fulfilled by all curves : 



(''"£+'^^^)"*=''- 



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



/cos«-^ +sin04-V0 = O. 

 \ dx dy) ^ 



But in a curve which fulfils the equation ^+ ^ =0, we have 



dx dy 



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

 branches is 



I (cos 0+ sin 6^)4-1 = ^- 

 \ dx ] 



Choose for the axis of a? a tangent to one of the branches at the multiple 

 point. Then it is evident that the preceding equation is satisfied by the In 

 values of % corresponding to the 2/ith roots of unity, that is to say, by 



6=0, J, (2«:ii>r. 



n n n 

 therefore the n branches make with each other 2w equal angles of ^. Q.E.D. 



