of Unipolar Induction. 417 



rotation whilst the magnet is at rest, the effect of the magnet 

 situated at c upon the element of the circuit As will be 



M.rRv(R 4- r cos u)Asdu 

 (R 2 + r 2 + 2R?*cosw)l ' 

 which will give therefore for the whole of the ring, 

 ^n/r a C 77 (R 2 +rll cos u)du 



i 



(R 2 + r 2 + 2Rrcosu)% 



The function under the sign of the integration becomes 

 equal to that of the preceding case if we replace r by R, and 

 R by r. The elliptic integrals would then also become equal 

 in consequence of this modification. The value of the total 

 induction will then be from that time 



2M? 



3. Let us suppose that R=10. For the case when the 

 magnet is in rotation and As at rest, the induction will be, 

 according to formula (8), = — 0'00316MtfAs. If, on the con- 

 trary, it is the magnet which is at rest, whilst As is in rotation, 

 the magnitude of the induction will be, according to formula 

 (9), equal to 0*6331 Mt'As. The first of these values is only 

 about 0*5 per cent, from the second. 



Let us suppose that R=20r. If it is the magnet that is in 

 rotation whilst As is at rest, we shall have, from formula (8), 

 a value of induction equal to —0*0004 MvAs. In conformity 

 with formula (9), the induction will amount to 0*3148 Mi'As. 

 For the case when the magnet is at rest and As in rotation, 

 the first of these values is not quite 0*13 per cent, of the 

 second. 



Let us suppose, lastly, that R = 2r. Then the induction 

 will be, according to formula (8), — 0*5416 MvAs ; and ac- 

 cording to formula (9) it would amount to 3*9132 MvAs. The 

 first of these numbers gives nearly 13*8 per cent, of the second. 



The result of the preceding investigation is that the values 

 per cent, increase in proportion as the distance between As and 

 the axis of the magnet diminishes ; and that in the last case, 

 where R = 2?*, the value per cent, has become considerable. We 

 must, however, remember that in the preceding calculations 

 we have only taken into consideration the magnetism of the 

 external layer of the magnet. But the magnet is equally 

 magnetic in the layers at a shorter distance from the axis, 

 and the inductive power of these layers may" be calculated in 

 a similar manner as the effect of the layer limited by the cir- 

 cumference of the magnet. As the values per cent, will 



