56 REPOKT— 1881. 



by M. Camille Jordan's law ; so that in this, as lias also been proved to 

 be the case for all the forms of the 10th and lower orders, this so-called 

 limit to the order is the actual highest order itself. 



The computation of the ground-forms to the binary quantic of the 

 11th order, on account of the immense length of the calculations which 

 it would involve, has not been undertaken. It is to be hoped that this 

 may some day be accomplished, as it is probable that the generating func- 

 tion for this case would, like that for the 7th order, reveal some pecu- 

 liar features not observable in the generating function for quantics, the 

 orders of which are multiples of 2, 3, or 5. 



Since the publication of the lists of ground- forms (calculated by aid 

 of the British Association grant) in the ' American Journal of Mathe- 

 matics,' Dr. Von Gall, of Mainz, has rendered a very important service 

 to algebraical science by applying the German method to the ascertain- 

 ment of the ground-forms of the binary quantic of the 8th order, and 

 his final results have been brought into perfect accordance with those 

 contained in the lists above referred to, witli the exception of one single 

 form of the deg-order 10'4, which he has not been able to decompose, but 

 which ought to be reducible if the fundamental postula e employed in 

 the English method is valid. It became, therefore, a matter of great 

 importance to demonstrate that no ground-form of such deg-order can 

 exist, for this will probably be the last occasion when it will be possible 

 to compare the results obtained by the two methods. 



Accordingly, Professor Sylvester, in conjunction with ]\Ir. Morgan 

 Jenkins, has calculated certain of the ground-forms for a particular func- 

 tion of the 8th order, and has therebj'- been able to demonstrate the 

 impossiblity of the existence of a ground-form of the deg-order in ques- 

 tion. This impossibility is made to depend ultimately on the fact that 

 the minor determinants of the 10th order belonging to a matrix 11 places 

 wide and 10 deep d© not all vanish, and the correctness of the figures 

 appearing in this matrix is demonstrated by showing that a certain de- 

 terminant of the 11th order, formed by adding on another line of figures 

 to this matrix, does vanish, which is practically effected by testing its 

 value in respect to various numerical moduli. The calculations have 

 been made independently by Mr. Sylvester and Mr. Jenkins, and concur 

 in proving (to the highest degree of moral certainty) the correctness of 

 the previous work. The non-existence of the doubtful ground-form may 

 accordingly be regarded as now placed beyond all reasonable doubt. 



A summary of the method employed, and the calculations to which 

 that method leads, has been sent for insertion in the ' Comptes Rendus ' 

 of the Institute of France, and the whole of the work Avill be reproduced 

 in detail in the forthcoming Number of the ' American Journal of Mathe- 

 matics.' 



