518 On a Theorem relating to t/ie Products of Sums of Squares 

 whence it is easy to see that 



< + «//' + V + ^/'+ • • • ={io'' + a^ + b^ + c^ . . . ) 

 {wl' + a^ + b^ + c^...) 



-\-2X\e^{ab^—ap){cdi— c^d) 

 + ^{ac— a^c) {dbi — d^b) 

 u'{adi — aid) {bc^ — b^)'] . 



where the summation extends to all the quadruplets formed 

 each by the combination of two duads such as ab and cd, or ao 

 and db, or ad and be, i. e. two duads, which, combined with the 

 same common letter (in the instances just mentioned e, or/, or ^), 

 enter as triplets into the system of quasi-equations — so that if 

 v=2"— 1, the number of quadruplets is 



l/v-lv-3\ 1 _ v(v-l)(v— 3) 

 2V~2~~Jrr3 - 24 



And the terms under the sign 2 will vanish identically if only 



but the relation ee' = tt' is of the same form as the equation 

 ee' = ^; hence if all the relations 



are satisfied, the terms under the sign S vanish, and we have 



which is thus shown to be true, upon the suppositions — 



1 . That the system of quasi-equations is such that 



being any two of its triplets with a common symbol e^, there 

 occur also in the system the triplets 



/o«o^o' fo^oK 



2. That for any two pairs of triplets, such as 



the product of the signs of the triplets of the first pair is equal 

 to the product of the signs of the triplets of the second pair. 



In the case of fifteen things a, b,c . . . the triplets may, as 

 appears from ]\Ir. Kirkman^s paper, be chosen so as to satisfy 

 the first condition ; but the second condition involves, as Mr. 

 Kirkman has shown, a contradiction ; and therefore the product 



