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VII. 



A 3-DIMENSIONAL COMPLEX A T AKIABLE. 

 By S. B. KELLEHER, MA. 



Bead January 11. Published September 25, 1915. 



This paper aims at giving geometrical interpretation to some properties of 

 the complex variable x + yd + zd 2 , where 8 3 = 1 but 1 + 6 + Q~ is not zero. 

 This variable has been already considered by de Morgan (Trans. Phil. Soc. 

 Cambridge, viii (1849), p. 241), and in another form has been classified 

 lb. (n = 3) by Study, Nach. Gott. 1889, p. 237. The modulus employed is one 

 which was used by de Morgan, and the results obtained show some analogy 

 with propositions concerning the ordinary complex variable. 



While the theorems concerning the differentiation and integration of 

 3-dimensional and a -dimensional variables are well known, it is possible that 

 some of the geometrical results obtained have not appeared in print before. 



I. If we call P + Q + R the modulus of the expression P + QO + RO 2 

 where 6 is dealt with as an ordinary algebraical quantity subject only to the 

 relation 6P = 1, but such that 1 + 6 + 6 2 is not zero, it follows easily that if 

 we take two such expressions as P + Q8 + Rft 1 



(a) the modulus of the sum = the sum of the moduli, 



(b) the modulus of the difference = the difference of the moduli, 



(c) the modulus of the product = the product of the moduli, 



(d) the modulus of the quotient = the quotient of the moduli. 



II. This expression P + Qd + BO 2 where P, Q, R are functions of three 

 independent variables x, y, z has a differential coefficient with respect to the 

 variable x + yd + zQ 2 provided 



dP_dQ_ dR 



dx dy dz 



(IP dQ dR 



'd~, ~ lz~ ~ die ' (A) 



dP _ dQ _ dR 



dz dx dy 



The sums, differences, products, and quotients of expressions which 



