44 MAJOR P. A. MACMAHON: MEMOIR ON THE 



Art. 130. Resuming the discussion, we proceed to eliminate </ and h and remark 

 that the second fraction may be omitted as contributing no term free from h. 

 Eliminating g from the first and g from the third, we have 



,, 



i Y Y 2V 2v v 2V v 3\ /i Y Y 2 Y Y 2 Y 2 Y Y 3 \ 



L AjAg A 4 A5A 8 AgAjQ J ^ 1 AjA 2 A5A<j A 7 A 9 Ajy J 



(lh X 3 X 6 X 7 X 10 ) ( 1 j- X 2 X 4 X 9 .A 



aV 2V 2V 2V 2V aV 2V 2V 2V 2V 

 Aj A 2 A 3 A 4 A 8 A B A 7 A 8 A 9 A t 



r '' V V 2V 2V V 2V V 3\ /l Y Y 2 Y Y 2 Y 2 



Aj A 3 A 4 A 5 A 8 A 9 A 10 ) (i A! A 2 A 5 A 6 A 7 



(1 h X 3 X 5 X 7 X 1(I ) ( 1 Y XiXtfXaXio j ( 1 j- 2 X 1 X 2 X 4 X )) X 8 X 9 Xi 



Art. 131. If the diagonal property associated with (j is alone to be satisfied in 

 addition to the row and column properties we may put h = I. Observe that the 

 second of the three fractions cannot now be omitted. Simplifying we obtain 



1 V 2V" 2V 2Y 2V 2V 2V 2V 2V 2V 6 

 AI A 2 A 3 A 4 A 5 AK A 7 A 8 A 9 Am 



( L XiXgXgXio) ( I X 2 X 4 X 9 Xn,) (l X 3 X 5 X 7 X 10 ) 



II V V 2V V 2V 2V V .1\ /I V Y 2Y 2Y Y 2V Y 

 . ^ i ^-1^-2 ^-5-^-6 A 7 A 9 A ll( ; ( I A t A 3 A 4 A 5 A 8 A 9 A 



Establishing the five ground products 



X 3 X 5 X 7 X UI , 



XY 2 Y Y 2 Y 2 Y Y :! 

 !A 2 A 5 A 8 A 7 A 9 A, , 



XV 2V 2V V 2V V :t 

 lA 3 A 4 A S A 8 A 9 A 10 



connected by the ground syzygy 



(X 1 X 6 X 8 X 10 ) :J (X 2 X 4 X 9 X 10 ) 2 (X 3 X 5 X 7 X 10 ) 2 

 = (X 1 X/X 5 X, 2 X 7 a X 9 X 10 3 ) (X 1 X/X 4 2 X 5 X/X 9 X 10 3 ) 

 corresponding to the fundamental squares 



100 010 001 

 001 100 010 

 010 001 100 



1.20 102 



^012 210 



201 021 



