On Continued Fractions in Quaternions. 117 



This reduces equation (11) to the following, 



which shows that the ratio of the effect of an electro-magnetic 

 engine, working by the induction of magnetism in a soft iron 

 bar, to the power expended, approaches unity as the velocity 

 increases without limit ; but at the same time the actual per- 

 formance diminishes indefinitely. 



(10.) The foregoing examples illustrate the method of apply- 

 ing the general law of the transformation of energy to some of 

 those forms with which we are most familiar, by interpreting the 

 symbols according to the properties of the kinds of energy under 

 consideration with which we are made acquainted by experiment. 

 The examples are necessarily of a simple kind; but I purpose .to 

 apply the same principles to determine the laws of more compli- 

 cated phsenomena. 



Glasgow, December 31, 1852. 



XIX. On Continued Fractions in Quaternions. J^y Sir William 

 Rowan Hamilton, LL.D., M.R.I.A., F.R. A. S.^c, Andrews' 

 Professor of Astronoimj in the University of Dublin, and Royal 

 Astronomer of Ireland^. 



[Continued from vol. iv. p. 303.] 



4. nnnOSE who have acquired some familiarity with the 



-1- interpretation of the results obtained by the Calculus 



of Quaternions, will have now little difficulty in seeing that the 



following geometrical theorems f are obtained from the consi- 



(8 \* 

 — — - j pQ, where ot,ff,pQfpM 



are real vectors, /3 being perpendicular to the other three, and 

 the condition cxf^ 4- 4/3"^ ,> being satisfied. 



Let c and d be two given points, and p an assumed point. 

 Join DP, and draw cq perpendicular thereto, and towards a 

 given hand, in the assumed plane cdp, so that the rectangle 

 CQ . dp may be equal to a given area. From the derived point 

 Q, as from a new assumed point, derive a new point r, by the 

 same rule of construction. Again conceive that s is derived from 

 R, and T from s, &c., by an indefinite repetition of the process. 

 Then, if the given area be less than half the square of the given 



* Communicated by the Author. 



t These theorems are taken from art. 665 of the author's (as yet un- 

 published) Lectures on Quaternions. 



