4-96 The Rev. T. P. Kirk man on Pluquaternions, 



Theorem: The -product of two sums of [2*(3A + A) + 8] 

 squares is always a sum of 



[(3kf + 3k • (2h + 5) + h ' (h + 7) + 8] squares, 



where k and h are positive or nothing, and h not > 2. 



Cor. The product of two sums of [2 • (3k -f- h ) + 8] squares is 

 always a sum of [2 • (3k + h) + 8] squares, if [4* (3k + h) + 12] 

 of the roots in the two factors are such as will satisfy [(3£) 2 

 + 3k(2h + 3) + h-(h + 5)] assignable conditions, viz. the condi- 

 tions $3 2 =0. 



In the conditions 33 2 = 0, the four quantities ww t aa t do not 

 occur: these are therefore arbitrary; and if the conditions 

 are satisfied by (4«— -4) of the given 4w numbers, we can make 



^=^ = 0, and — can always be so taken as to cause one of 



the quantities (w tl a n b n c tl , &c.) to vanish ; so that we then shall 

 have the product of two sums of (2^—1) squares equal to a 

 sum of (2n — 1) squares. 



A glance at the functions BX^ . . . BjC, . . . , &c. will suggest 

 a simple relation between the numbers (bc.r) and (^...r,), 

 by which the conditions 33 2 =0 are all at once satisfied; and 

 as a case of the preceding corollary, we have the theorem, 



= a 2 + a / 2 +a 2 2 -f-...+a* n _ 1 ; 



whatever be the numbers in the first member of this equation. 



We have proved, that, whenever the real quantities in two 

 pluquaternions of (2«—l) imaginaries are so related among 

 each other that the product of the two functions is also a 

 pluquaternion, the product of the 2n squares (w 2 + « 2 + ... +r 2 ) 

 into the 2n squares (wf + af + bf-t- ... + rj*) is always equal to 

 the sum of the 2n squares (wf + af- + bf + ... + r y 2 ) ; and that 

 the product of any In squares into a sum of any other In 

 squares can always be reduced to a given even number of 

 squares. The conditions that a sum of ten squares into a 

 sum of ten squares should give a product of ten squares, we 

 find to be not more than six, among sixteen of the roots in the 

 factors ; while the number of conditions required that the like 

 should hold for sums of twelve squares, we have ascertained 

 to be at the most fourteen, among twenty of the twenty-four 

 quantities. 



In the equation 



J *V, 2 = J% 2 -Q„ / R+ RQ_„, -RR, 



