63: 



TRANS. ST. LOUIS ACAD. SCIENCE. 



lie on the surfaces of two cylinders whose traces upon the plane 

 w, r, T are the two circles dd of Fig. i . When a =^ o ^ the 

 maximum and minimum points are at A and A\ the values of 

 I^' being ■ 



^' ,n' = d= 



' V* 



If r be the perpendicular from m to any other line determined 

 by the angle a so that 



r =z r' cos a , 



the maximum J^' along this line will be similarly 



Hence, 



^'. = ± 





'^ § V* 



The points representing the maximum and minimum J^' will 

 therefore lie in a kind of spiral around the two cylinders. If we 

 suppose a to vary from o to -|- 90, the maximum will move 

 through the arc A O m, varying in value from jF'm' to co , 



