304 



Mr DAVIES on the Equations of Loci 



lution of 6 carries the point to the opposite pole L. A third brings it back 

 to the equator at Q ; and a fourth back to P. The general figure of the 

 curve will be well understood from Fig. 13., keeping in mind that all cuspi- 



dated points in the circle PELQ, as F, G, H, I are indicative of the 

 dotted and traced branches whose projections meet in those points, forming 

 continuous portions of the curve. 



4. If m and n be commensurable, the branches of the spirals traced out 

 will return in the same order, and each coalesce with its corresponding one, 

 after 6 has attained the value QmnTr; but if they be incommensurable, there 

 can never be any reduplication of the spirals traced out. 



XVIII. 



The rectification of these spirals can always be effected by arcs of an 

 ellipse, but never by any simpler functions, 



For, in this case, we have - 0, and d6 = d<b. Hence, the ele- 



n ' n 



ment of the arc becomes 



