320 REV. T. p. KIRKMAN ON THE THEORTf OF 



t'r I'n rn 



Pm Pm Pm 



that we can remove the exponent from the denominators 

 of Qi- •Q,„, if we either add 4-2 to the exponent of the 

 factor over p^, or subtract 2-1 from that exponent, with- 

 out changing at all the derivants Qj • • GI5 . After such 

 modification the system of subindices and exponents read 

 in Qi • • Q5J neglecting p, will be represented by 



1 23 



2 3^^=^ 

 3^1 2^ 

 1 3^2^ 



{9") 



f2 1' 



2 1 3. 

 which differs from {g') only in having every exponent of 

 the third vertical row augmented by 2, or, what is the 

 same thing, diminished by 1. 



Is now g" a group, as well as g'? If it is, it will be 

 unaltered, if we multiply it by any of its substitutions. 

 We ought to have 



{2fi^)g"=g". 



The rule of Art. 3 is still our guide; and this gives us 

 23^1^ for our first line. What then is 

 23^1^. 23"^^? 



Our rule bids us write 2 for 1 ; we can then for 1^ write 

 nothing but 2^. Our rule bids us next write 3'^ for 2, 

 which is correct. We are next to write 1^ for 3 ; then for 

 3^+^ we must write 1^+^= 1. This gives the product 





{2fir= 



--Sh2\ 



which it ought to be. 







The next step is 







23 



fl^.fll'': 



= 123, 



and thus we easily satisfy ourselves that 2'^i^-g" is merely 

 g" in a different order. 

 If now in g" we put 



