49S Sir W. R. Hamilton on some 



by replacing the internal wheels by others, which, instead of 

 bearing electro-magnets, are surrounded by a ring of soft iron, 

 which forms the armature j the moveable portion is thus rendered 

 lighter, and the teeth are rendered unnecessary. It is this 

 modification of the machine that we have seen in action. 



The circular electro-magnets of M. Nickles will here find an 

 interesting application ; and, at our suggestion, M. Marie pro- 

 poses to make some experiments with this addition, which will 

 enable him to augment the power of the machine without 

 increasing the expense. 



The construction of the machine is somewhat aflFected by 

 the inexperience of the maker, so that it required a battery of 

 twenty-four Bunsen^s elements to produce -^ horse-power. 

 But, according to M. Marie's calculations, one of the same 

 energy, or perhaps even one of less intensity, would be suffi- 

 cient, with a machine of large size, to produce 300 times the 

 power, seeing that the friction would not increase in the same 

 proportion as the force of the machine ; the means of electrical 

 communication not being changed, and the power produced by 

 the attraction of the magnets being capable of multiplication in 

 a great degree, by making use of electro-magnets formed of 

 lai'ge cylinders of soft iron. The model was constructed with 

 a view to show the relations between the effect calculated from 

 the magnetic force developed in the electro-magnet and the actual 

 force produced. The proportion was as 4 to 3, which is already 

 a very close approximation, considering the numerous imperfec- 

 tions resulting; from the bad construction of the machine. 



LXXVII. On some Extensions of Quaternions*. 5y Sir William 

 RoAVAN Hamilton, LL.D., M.R.I. A., F.R.A.S., Correspond- 

 ing Member of the French Institute, Hon. or Corr. Member of 

 several other Scientific Societies in British and Foreign Coun- 

 tries, Andrews' Professor of Astronomy in the University of 

 Dublin, and Royal Astronomer of Ireland-^. 



[1.] /CONCEIVE that in the polynomial expressions, 



P' =to*'o + ii*'i+ ••+*««'» =Sta?', ^ . . (1) 



the symbols x^. . . a?„, which we shall call the constituents of the 

 polynome P, and in like manner that the constituents Xq . . a?'„ 



* See the work entitled, " Lectures on Quaternions," by the present 

 writer. (Hodges and Smith, Dublin, 1853.) 

 t Communicated by the Author. 



