96 Prof. Cayley. On the [.June 19, 



not greater than 1, and in any such curve consider as extending 

 from 0=0 to 0=+ - ^ : writing this last valne=;f 7, we have 



K 



k= . - ; if 7< TT, that is, k< , , the curve is a mere arc cut- 



ting at right angles (at the distance r=k from the centre) the 

 meridian 0=0, and extending itself out on each side to meet the unit- 

 circle in the points 0=7, 0= 7 respectively ; in the case 7=T, that 



is, fc= , - =, the two points 0=7 come together at the point 



V 1 + JT 2 



0=7r, or the curve becomes a loop; and for larger values, &= / = 

 to / -^= 0) we have the two branches crossing each other on the 



V 1+4JT 2 



k 

 meridian 0=?r at the distance r= , - - from the centre and then 



VI - K-9T 2 



extending themselves in the opposite semicircles, so as to meet the 

 unit-circle at the points 0= + 7. And we have thus another critical 



value fc= / , for which the two branches having thus crossed 



V 1+4JT 2 



each other come to unite themselves at the point 0(=2jr)=0 of the 

 unit-circle ; and in like manner the critical values 



V* 



, &c. : for a value of k between such limits the branch is a 



spiral having a determinate number of convolutions, and the two 

 branches cross each other always on the radii 0=0 and 0=jr 

 respectively. 



21. Let YT denote the inclination of the radius vector to the normal, 

 or what is the same thing, that of the element of the circular arc to 



the tangent; wo have tan yr=_!l, and -?= ^,=^0, that is, 



r<i0 rd<f> 1 &*0* 



tan y<-=r 2 . At the intersection with the unit section r=l, and there- 

 fore tan^ r =0; moreover putting =cos K, so that the equation 



of the curve now is r*= 1 * o , then for r=l we have 0=tan *; 



and hence at the intersection with the unit-circle V r=K ; that is as Tc 

 decreases from &=1, or k increases from fc=0, the angle at which 

 each curve cuts the unit-circle is always =, and thus this angle 



continually increases from =0; for k= , ==cosy, and therefore 



tan *=*-, we have r=72 20' nearly, the complement hereof 17 40' is 



