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



apply to ©'„,/. On the contrary, (vg.) and (vg.) will be clearly 

 applicable to ©"w,/j f^n^l "ot ^o ©'„,/• 

 If we take ^j = ^ = 0, we shall have 



0'„^^= i. (.r„ + .^".r^ + i3^^■y + .^"^rs + i'^T,), 



%\j= —{x, + i"a^p + .2«^^ + ,3«a:s + i*"^,), 



>' W 



5 



But having decided upon thus fixing the meanings of ®'w,/ 

 and ®"n,fi we must be careful, in evolving particular forms 

 either of @M,/(ab) (cd) . . or of (@„^y) (^a b) (c d) . . , to attend 

 to the equations (x.) as well as to (v.) or (w.). 

 There may also subsist 



and 



®V/(«.) = (®V) (7.0,1 ,. 



®«,/(/3e) being equal to ®n,f(y}y ^^^^ ®m,/(«/3) ^^ ^n,f(yjy 



Section VIII. 

 39. Again, if we examine the equation 



on designating, for the moment, 6„ (/3 ^ by I„, we shall per- 

 ceive that 



Jo + »4I, + ,31^ + ,21^ ^ J^ = ^^ ^ e^ _^ ,(H)^ + ,2@^ + ,30^ + ,40^^ 

 lo + ^% + i% + ilg + .^I, = ^y = ©0 + .2@^ + ,4@^ ^. ,0^ + ,30^^ 



Io+«^Ii + J2 + .% + .% = ^5 = @, + ,S0^ + ,©2 4-,i0^+,20^, 



Io + «Ii + .%+l% + »4l4 = ^. =:0O+l'*@, + *^®2+''@3+»®4; 



from which will result 



5©„ = (3 + ,2« + .3«) 1^^ ^ |i + 2 02« + ,3.)} I2^^ 



+ {1 +2(.« + ,4«)} l3„ + (3 + ," + *4«)l4,^; 



which connects one system with the other. 



Hence if we denote 



.S + |2 + ,3, 1+2 (»2 + ,3)^ 1 + 2 (i + J-*), 3 + I + A 

 by a, b, c, </, 



respectively, we shall have 



