and Homoid Products of Sums qfn Squares. 505 

 from which can be deduced, as before, if 



whose modulus is U(,)j 



Q--Qf»-«. ... Q/Q^Q^+Q^ftw^^+ttw + e, 



=(Gu i --Ifc (l) -0 i ; 

 where ( . is a function of imaginaries of a degree higher than 

 the nth. If we can now satisfy ourselves that all the (n-\- l)plets 

 of these two products can be reduced to multiplets of lower 

 degrees, or that 0j = O, we shall obtain by multiplication, as 

 before, 



or the product of (n+l) sums, and consequently of (n + r) 

 sums, each of any (2m + 8) squares, is always reducible to a 

 constant number of squares, which is known, as soon as we 

 can assign the constant number of terms, if such a number 

 exists, to which it is possible to reduce the product of (n + r) 

 pluquaternions of the order i( = 6n + 1 or 6n~ -3). 



We now proceed to inquire, not without interest in the re- 

 sult of our investigation, whether such a thing exists as the 

 complete pluquaternion product of (n + r) pluquaternions of i 

 imaginaries, which is always of one form (<St 4 + !&(,•)), for every 

 value of r. The subindex (*) may serve to characterize this 

 product. 



I flatter myself with the hope of establishing the proposi- 

 tions which follow: — [« not < 0, r not < 0]. 



The product of [n + r) pluquaternions of 6n — 3 imaginaries 

 is always reducible to 2 2 " terms. 



The product of {n + r) pluquaternions of (6n — 1) imaginaries 

 is always reducible to 2 2n+1 terms. 



The product of (n + r) pluquaternions of (6w + 1 ) imaginaries 

 is always reducible to 2 2n+1 terms. 



Of the truth of these propositions, once established, the in- 

 evitable consequences will be the following : — 



The product of (n + r) sums, each of any (6n — 2) squares, is 

 reducible to a sum of 2 2w squares. 



The product of (n + r) sums, each of any 6n squares, is redu- 

 cible to a sum of2' n+1 squares. 



The product of (n + r) sums, each of any (6n + 2) squares, is 

 reducible to a sum qf2 2n+1 squares. 



Phil. Mag. S. 3. No. 225. Suppl. Vol. 33. 2 L 



