ACTION OF A FIELD ON A MAGNETIC SHELL. 



327 



</Q being the increase of the flow of force which traverses the 

 negative face of the shell. The work dT of the magnetic forces 

 being equal and of opposite sign to dW, we have 



As the form of the shell is a matter of indifference, we may suppose 

 that it forms part of a continuous surface S, passing through the 

 positions C and C\ which the edge occupies, and that this only 

 makes it glide on the surface. 



The work of the magnetic forces is proportional to the excess 

 of the flow of force which traverses the surface bounded by the 

 edge G! over that which traverses the surface bounded by the 

 edge C. The flow of force relative to the portion common to 

 the two shells disappears by difference, so that calling q and q' the 

 flows which traverse the spindles AMB and AM'B, we have 



Let ab be an element of the first contour, a^ its new position 

 after the displacement, F the force of the field at this point. To 

 obtain the part dq of the flow relative to the displacement, we must 

 multiply the force F, by the projection of the parallelogram, abb^ 

 which this element has described, on a plane perpendicular to this 

 force. 



Fig. 81. 



Finally, in order the better to see the geometrical signification 

 of this product, let us imagine an observer laying along the curve C 

 so that, looking at the shell, he has the negative face on his right 

 hand. The positive direction of the arcs is that of a moving body 

 which goes from the feet to the head of the observer. Let us take 



