560 Mr. G. B. Jerrard's Reflections on the Resolution 



we shall ihen, by making the following substitutions with /5 

 and e. 



find the six expressions of whichy, is susceptible, 



i/2 + i/5 - (' + ''^) yi» 3/3 + 5/4 - (' + »^) 3/1. 



J/4 + J/6 - (' + *'*)i/i» ^ 5^2 + i/3 - (' + »'') j/i ; 

 or, if we designate the expressions in the first of these columns 

 by /j, ky, /i, respectively, 



hi c /j, 



each expression in one column being the complement of the 

 corresponding expression in the other. 



By applying (? 9 4 k ) ^-o ^^ch equation in the system (d.), 



we obtain i^ = along with four equations which, from ana- 

 logy, we shall indicate by ic^—O^ 4=0, i^ — O, 4=0, taken in 

 order. 



Again, by applying ("3245) to each equation in the 



same system, we obtain Atj = followed by four equations, 

 which abstractedly of the order in which they present them- 

 selves, will be indicated by kc^= 0, y{-g=o, ^4 = 0, k^= \ 

 the index of k being made to correspond with the index of 

 the term multiplied by a,. 



Lastly, by applying (i^l^s)* ^^ obtain /^ = 0, and 



/2=0, Z3=0, /4=0, 4=0. 



Now, from the theorem (h.) there will result 



V(2?)e.?) = (fv)(2?)n; 



v(32)0i?) = (cv)C?X, 



V<t3)(25) = (c/e-)(^?X' 

 the second members of which will, if b'= b" = b"'= 1, re- 

 duce themselves to c^, c^j, c/j. 



If then we observe that generally (10.) 



