456 The Rev. T. P. Kirkman on Pluquaternions, 



where the similar brackets have all the same sign. Any other 

 function, as q tli will have the form, if we take the Jirst of our 

 two types, 



1u = g»i + Xb Qi ± i ra i - ar i\ ± { so i ~ os i ~ l Pi ~Pi } ? 



but which of these Jour shall be q ni can only be determined 

 when the system of seven triplets, to which q belongs, shall 

 have been exactly defined. We may thus have any number 

 of complete systems of seven, having one common imaginary. 

 It is obvious that the reasoning (C.) applies to every duad 

 coefficient in R 4 , R 5 , R 6 ; and it is easily shown to be true that 

 the property 



m o n o -\-n Q m Q -0 



belongs to every pair m n in the product of any two pluqua- 

 ternions, whether that product be a pluquaternion or not, pro- 

 vided that that product be formed consistently with a system 

 of congruous triplets. And all this is the inevitable conse- 

 quence of our first definitions of the imaginaries, and of the 

 supposition that Q tt and Q a/ have, if possible, a product of the 

 same form. 



The preceding analysis shows that 

 R^'l 

 R 2 =6'2 + 2 

 R 3 =6'3 

 R 4 =6-4 + G-l 

 R 5 =6'5 + 6-2 + 2 

 R 6 =6-6 + 6'3 

 R 7 = 6-7 + 6-4. + 6'l 

 R 3 = 6'8 + 6-5 + 6-2 + 2, 

 &c. &c, 



the law being obvious. 



Generally, if we form the product 



Qa Q a/ =Q«„+Rm 



of two pluquaternions of (2m + 7) imaginaries, the number of 

 terms in the condition-function R TO , if ?n = 3k + h (k not <0; 

 A not <0,not>2),is always [(3kf + 3/r (2h + 3) +h' (A + 5)]. 

 Since each of the imaginary duads in R m must of necessity 

 appear, either by itself or by its equivalent, in the product of 

 Q a and Q a/ , it is impossible that this product should contain 

 fewer than [(3&) 9 + 8k ' (2* + 5) + h(k + 7) + 8] terms; and if 



