38 BEPOBT— 1880. 



From these tables it appears that although the deviations are less for 

 Legendre's formula than for the li x formula, the former increase in a 

 more rapid ratio than the latter. As Legendre's formula contains a 

 disposable constant, chosen so that the values given by the formula might 

 represent well the results of the enumerations for comparatively small 

 values of x, it is to be expected the deviations would for some time be less 

 than in the case of the logarithm integral formula, in obtaining which x is 

 is supposed to be very large. 



The portion of the former of these two tables up to 4,000,000 has 

 appeared in a paper ' On the value of the constant in Legendre's formula 

 for the number of primes inferior to a given number,' ^ but the extension 

 to 5,000,000 is new. This paper also contains comparisons between the 

 numbers of primes counted and those given by the formulas : 



log 33—1 

 and X 



log x—1- 



log X 



up to 4,000,000. These have also been extended to 5,000,000 ; but it 

 seems scarcely worth while to give the tables here, as the extension 

 amounts to only one million. 



Report of the Committee, consisting of Professor Sylvestek {Chair- 

 man), Professor Cayley, a7id Professor Salmon, appointed for 

 the purpose of calculating Tables of the Fxindaraental Invariants 

 of Algebraic Forms. 



In consequence of the academical engagements of Mr. (now Dr.) F. 

 Franklin, the trained and skilled assistant in the computation of the 

 tables, only a small portion (81. 5s.) of the SOL granted by the Associa- 

 tion has been expended. 



With this sum the tables for the generating functions and ground- 

 forms of all single quantics, up to the lOfch order inclusive, have been 

 corrected and completed, and the tables relating to binary systems of 

 quantics for all combinations of orders up to the 4th inclusive, re- 

 calculated. The results have been published in extenso in the ' American 

 Journal of Mathematics.' 



This revision has led to the discovery that two of the forms included 

 in the table of ground-forms for a pair of cubics previously accepted as 

 correct arc composite forms, and should be omitted from the catalogue. 



The table affected with this error had been calculated by the German 

 mathematicians after Gordan's, and by Mr. Sylvester after an entirely 

 different method, and the results were in perfect but fallacious accord. 



The German method, it may be stated, never offers a complete 

 guarantee against the occurrence of an error of this nature ; its per- 



' Proceedings of the Camhridge PMlosojMcal Society, vol. iii. pt. vii, pp. 295-308 

 December 8, 1879). 



