EEPOETS 



ON THE 



STATE OF SCIENCE. 



Report of the Comniittee, consisting of Professor Sylvester, Pro- 

 fessor Cayley, a7id the Eev. Gteorge Salmon, D.D., appointed in 

 connection with the Calculation of Tables of Fundamental In- 

 variants of Binary Quantics. 



It has been thought advisable to extend the calculation of tables of 

 invai-iants (proper) to ternary systems of binary quantics, and the follow- 

 ing systems have been selected for the purpose. 



Two quadrics and a quartic ; a quadric and two quartics ; three 

 quartics : a quadi'ic, cubic, and quartic : say the systems (2) (2) (4) ; 

 (2) (4) (4); (4) (4) (4); (2) (3) (4). 



By far the most considerable amount of the work belongs to the cal- 

 culations connected with the last-named system. 



The entire work covers 52 sheets of paper of dimensions 31 in. by 23 

 in., divided into 93x69, i.e. 6417 ^rd-inch squares, each enclosing one 

 numerical coefficient ; the total number of such spaces, of which only a 

 moderately small fraction remains vacant, being accordingly 333,684. 



The work of compilation was actually performed by Messrs. Healy 

 and Durfee (fellows), under the able superintendence of Dr. F. Franklin 

 (associate) of the Johns Hopkins University — who is of opinion that 

 at least nine-tenths of the labour and time that would otherwise have been 

 required for the calculations has been saved by the method of operation 

 above indicated : a method called by its presumable inventor or originator, 

 Professor Sylvester, the method of cage-work or occlusion, which has 

 also been found by Dr. Franklin applicable with very considerable 

 advantage to certain operose astronomical computations with which he 

 has been entrusted in connection with one of the public departments in 

 Washington. 



The leading results will be published in the ' American Journal of 

 Mathematics ' in a form similar to that of the tables of like kind, calcu- 

 lated at the expense of the British Association, which have already 

 appeared there. 



