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



where tt)^ . . . r n are certain functions of the arbitrary numbers 

 lew, aa,... rr r 



In this last equation, it is plainly implied, that every duad, 

 a Q b oi that can be made with two imaginaries, is a linear func- 

 tion of those units ; i. e. we must have 



a b =Aa + Bb +Cc + .... +Rr Q ; 

 whence, since « 2 = —1 =0% 



-b Q = - A + Ba b + Ca c + + R<V > 



-a =Aa Q b o -B + Cc o b o + + Rr ft . 



Therefore A = = B j and 

 -b o *=l = Cb o a c + Db a o d + Eb o o o e o + ... + Rb a o r o . 



As b Qi a oi c Q , &c. are pure imaginaries, we cannot suppose 

 that any term of the last expression, as Cb o a Q c 0i is of the form 

 V+ W, where V is real and W imaginary. Some one or 

 more, then, of these terms must be real ; say the first three 

 of them. We have then 



C* « c =C', 

 D6 « rf =D', 



Ebae = 1-C'-D'. 



It follows that 



C _D 



C' c °~ D' ** 



which is impossible by hypothesis. There can then be only 

 one of the terms real in the value of — i' 2 , and the equation 

 before us is of the form 



-b*=l=Cb a c , 



or assuming C= + 1, 



1 = ±&o«o c o; 

 which gives 



&o= + «o C o> <*cA=± C o> 

 as the form of the linear function in question, equivalent to 

 any duad a Q b Q or a c Q . 



One condition, therefore, of the possibility of the equation 



Q Q =Q 



is, that all the imaginaries can be combined into triplets, of which 

 no two shall have a duad in common ; since if b Q a Q c Q BX\<\d Q a Q c Q 

 were both real, there would exist a linear relation between b Q 

 and d Qi contrary to hypothesis. The number (2w— 1) of the 

 imaginaries is hereby restricted to the two forms 6m + 1 and 

 6?n + 3, in the case of either of which the triplets can be formed 



