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



These curious equations of the form w = ('w/w ll + a l a ll + . . -f 

 r / r //)*i^r 2 ' as we 'l as pV 2 /—/* 2 //* are familiar to those who are 

 acquainted with the masterly Theory of Quaternions invented 

 by Sir William R. Hamilton, by whom they have all been 

 discovered several years ago, for the case of three imaginaries ; 

 and as the process of rinding w, «, 6, c in terms of w, w n a { a ip 

 &c. is called by him Division of Quaternions, so may the ex- 

 tension of the process to cases of more than three imaginaries 

 be called Division of Pluquaternions. 



We have here arrived at these remarkable results by purely 

 a priori reasoning; and we accept them at once, without ask- 

 ing for the application of any test, or troubling ourselves about 

 the multitude of vanishing terms. 



In our first definitions of the imaginaries, and in the sup- 

 position that, if possible, the product of two pluquaternions 

 shall be a pluquaternion, irrespective of all limitations upon 

 their real quantities, we have found it to be of necessity im- 

 plied, that the form of that product should be in general, for 

 (2w -f 7) imaginaries, 



Qa Qa, = Qa„ + RwJ 



and, from thecondition that this product is congruously formed, 

 (a condition imposed by our first supposition), no matter in 

 which of the various practicable ways the multiplication is 

 consistently effected, — before we have examined, before we 

 have constructed, before we have even learned how to construct, 

 the functions w lt a u b ip &c. — we conclude inevitably, that 



and that 



a>= («>,«>„ + «,«„ + . . . +*y>,- 2 . 



After what has been delivered concerning the different forms 

 of the product of two pluquaternions, whose constituents are 

 all given positive numbers, it becomes an easy problem in 

 combinations to find the number of ways in which the sum of 

 squares, ju^ + IS 2 , can be assigned equal to the product ft 2 ^ 2 . 

 For instance, ju, /; 2 , the sum of the squares of the constituents 

 of Q fl// , any one of the 480 forms of the product of two given 

 biquaternions Q a and Q 0/ , whose constituents are (wa..g), 

 {'w l a l . . gy), all of determined sigtis, is equal to (w 2 + a 2 + . . -fg 2 ) 

 (tt) / 2 + a / ' 2 + ..+g / 2 )=/x- 2 j«. / 2 . If now in all the functions wyz^...^, 

 the eight roots in j«, y 2 , we change the sign of any one or more 

 of the sixteen numbers w...g to^V.gp we shall, unless we 

 change the signs of all the sixteen together, obtain in every 

 case a different set of roots w u a n . . . g n ; and, except in a small 

 number of thecases,adifferentsetofsquaresw // 2 + # / , 2 + ...+g ;/ 2 ; 



