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LXIV. On certain Functions resembling Quaternions, and on 

 a new Imaginary in Algebra. By James Cockle, Esq., 

 M.A., Barrister- at- Law. In a Letter to T. S. Davies, Esq., 

 F.R.S.L.$E.$c* 



To Richard Taylor, Esq., F.S.A. $c. 



My DEAR Sir, Royal Military Academy, Woolwich, 



October 27, 1848. 



I BELIEVE that Mr. Horner's attention was first directed 

 by me to the consideration of what he has happily termed 

 " congeneric surd equations." This was done in consequence 

 of a difficulty that I had found in discussing such an equation 

 which was accidentally brought before me ; a difficulty which, 

 from the friendship that had long subsisted between us, I very 

 naturally referred to him. His reply was printed in the Phi- 

 losophical Magazine about a dozen years ago; and it has 

 received proper attention from those algebraists who look 

 with due care to the fundamental principles of their science. 



Almost every new difficulty in algebra is the precursor of 

 more extended views respecting its principles ; and often out 

 of a case which may, at first sight, appear to be little more than 

 a conundrum, very important improvements in our general 

 theories often arise. All efforts to generalize our views, so as 

 to include the " conundrum," deserve, therefore, attentive con- 

 sideration ; and for this reason I send you for publication the 

 accompanying letter, which I have just received from your 

 correspondent and my friend, Mr. James Cockle, of the 

 Middle Temple. 



I hope you will be able to give Mr. Cockle's letter an early 

 place in your valuable Magazine, as it contains much that is 

 suggestive of inquiry. 



I remain, 



Yours very truly, 

 T. S. Davies. 



2 Church -Yard Court, Temple, 

 My DEAR Sir, October 26, 1848. 



YOU are aware that the new algebraical symbol, which I 

 have recently discovered, was suggested to my mind by re- 

 flecting on the structure of congeneric surd equations — a sub- 

 ject on which I have lately been much engaged. Did I require 

 any excuse for throwing the present investigations into the 

 form of a letter to you, I should find it in the fact that the 

 remarks of Horner on congenerics were put into a similar 

 shape. (See Phil. Mag., S. 3, vol. viii. p. 43-50.) 

 * Communicated by Mr. Davies. 



