354 REV. T. p. KIRKMAN ON THE THEORY OF 



ment distinctes, et appartiennent toutes au systeme con- 

 jugue : ce sont done les substitutions i^P^Q,-- ecrites 

 dans un autre ordre. D'un autre cote toute autre substi- 

 tution, alterant essentiellement les facteurs alterera le 

 produit." 



Let us try this rule on the simple group 

 ahcd 1234 



bcda , . -, . 2'?4i 



, , which IS 

 caab 3412 



dabc 4123. 



We may take any asymmetric function of the letters 

 abed. Take, as directed, the function 



ad-\-bc. 

 The product which we are instructed to form is 



FiFpFQ^^i = {ad + be) {ba + cd) {eb + da) {de + ab) . 

 It is perfectly true that this is unchanged by any sub- 

 stitution of the group. Let us operate on it with the sub- 

 stitution 



adeb 



abed' 

 which is not in the group, and which therefore, by the last 

 words quoted, ought to alter the product. We obtain for 



result, 



{ab + de) {da + eb) {ed+ba) {be + ad), 



which is the same product still. In the same manner we 

 shall find that it is unaltered by any of the three substitu- 

 tions, 



deba ebad bade 

 abed' abed abed' 

 none of which is in the group, and which by the rule 

 given ought all to alter the product. 



This French rule will be found equally misleading, 

 when applied to innumerable other groups, both simple 

 and complex. 



It would not be quite fair to lay this error to the charge 

 of the author quoted ; for he does not pretend to give any 



