108 MR. JAMES COCKLE : SUPPLEMENTARY 



V. — Supplementary Researches in the Higher Algebra. 



By James Cockle, M.A., F.R.A.S., F.C.P.S., 



Barrister-at-Law. 



Read November 29th, 1859. 



§ 33- 

 I am desirous of submitting to the Society a Supplement 

 to the " Researches " which I have had the honour of lay- 

 ing before it, and which appeared in the last volume of 

 its Memoirs. And, first, the relations 



a + d=g, a^-\-d'^=ig, b^ + c^= 10, 

 ab + cd= - {ac + bd) = ^, (f){i) - <f){i^) = p 

 may conveniently be added to those given in § 2, and it 

 will also be convenient to establish the notation 



§ 34- 

 We may pass in two ways from v^iwz + xy) to U,., and 

 either cyclically or by forming (epimetric) functions into 

 which each square and product enters once and once only, 

 and which contain no cube or higher power. In the 

 former case the operand is subjected to the processes 

 which Mr. Harley denotes by S'l and S'3 : in the latter, 

 using the notation of the theory of interchanges,* we are 

 led by X to Ui and UjC^f). And we may write 



* On this theory see Mr, Jerrard's Essay, &c. ; his Reflections, &c. 

 {Phil. Mag. June 1845) ; my Observations, &c. {Ihid. May 1857) ; &c. 



