218 



Trans. Acad. Sci. of St. Louis. 



in which a is the special value of co for which co' = 0. Com- 

 bining the last two equations gives 



ra ( 1 — cos co ) — m' (1 — cos co' ) = m ( 1 — cos a ) . . ( 24 ) . 



This is the equation of a line of force. Its direction at 

 A makes an angle a with AX. When A and A' have charges 

 of like sign the equation of a line of force is 



m ( 1 — cos co) + m' (1 — cos co') = in ( 1 — cos a). .(25). 



Equations (24) and (25) were obtained from electrical 

 considerations. In what follows it will be shown how they 

 can be obtained from o-eometrical considerations. 



Fig. 7. 



S ]E X 



In Fig. 7 suppose A and A to be the traces of two axes of 

 rotation, each perpendicular to the plane of the paper. AP 

 and A'P are two lines in the plane of the paper, which rotate 

 in such a manner that the versines of the angles PAX and 

 PAX change at uniform rates. This can be brought about 

 mechanically by making AP and AP the centre lines or 

 axes of two equal cranks AB and AB\ each of which is 

 driven by a separate slotted crosshead whose slot is perpen- 

 dicular to, and whose motion is parallel to, A AX. The cross- 



