NIPIIEU — PROPERTIES OF A FIELD OF FORCE. 623 



The minimum will move from A' to »«, varying in value from 

 — I^'m' to — CO . Similarly a change in a from o° to — 90° 

 we shall have symmetrical changes through the other semicir- 

 cles, A, d, m. 



These spirals each project on the diametral planes A m and 

 A' m of the cylinders as branches of equilateral hyperbolcE the 

 asymptotes of which are the lines A m and A' /;/, and the verti- 

 cal line through ///. The opposite branches of the hyperbolaj 

 would also correspond to the case where in is negative. 



If v be an abscissa of one of these hyperbolae, measured from 

 A as an origin, and k the angle 0mA, then the equation of a 

 right projection on the diametral plane is 



r' F' , 



F', 



^ V 2 sin k X 



The spirals project on a plane at right angles to A ?n in a 

 fourth degree curve somewhat resembling a section of a Florence 

 flask with an infinitely long, pointed neck. The equation of this 

 cvu've is 



r"^ { F',„~ r F',n,-^ - 1 



y- = 



_ r 



2 sin'^ ^IF'„ lF',n J J 





where y is the projection ordinate corresponding to x in the 

 previous equation. The maximum ordinate y of this curve is 

 found to correspond to a value F'„, = 2 F' ,„> , showing that the 

 values F'^ at points 90" from A A' around the cylinders are 

 twice the values at A and A ' . 



