1878.] 



Prof. Sylvester on Binary Quantics. 



11 



measures taken at Moncalieri, by P. Denza, will most probably appear 

 shortly in the publications of that Observatory. 



In conclusion I will subjoin in a single table the mean results for all 

 the elements of terrestrial magnetism at the different stations. 



Table VIII. 



Station. 



Dip. 



Annual 

 amount 

 of 



Sec. var. 



Hori- 

 zontal 

 force. 



Total 

 force. 



Annual 

 amount 

 of 



Sec. var. 



Declination. 



Annual 

 amount 

 of 



Sec. var. 



Cape Town. 



O III 



-56 2 22 



-4-69 



4 -3098 



7 7152 



+ -0047 



O III 



29 58 33 W. 



+ 1-68 



Colaba .... 



+ 19 14 40 



+ 1-56 



8 -0853 



8-5633 



+ 0-0043 



56 E. 



+ 1 - 75 



Bombay. . . . 



+ 20 6 46 





8 -0354 



8 -5572 





55 36 E. 







+ 5 28 9 



+ 0-68 



7 "6253 



7 -6602 





2 19 39 W. 





Port Said . . 



+ 42 31 54 





6 -4056 



8 '6926 





5 9 49 „ 





Malta 



+ 51 36 42 



-3-49 



5 -6389 



9 -0805 



-0 -0052 



12 8 25 „ 



-4-76 



Palermo. . . . 



+ 54 18 8 





5 '4005 



9-2551 





11 15 33 „ 





Naples .... 



+ 57 23 



-3-45 



5 -1451 



9 '4484 



-0-0033 



U 31 22 „ 







+ 58 50 hi 



-2-09 



4 -9207 



9-5235 



-0-0052 



12 16 33 „ 



-5-95 



Florence . . 



+ 60 13 



-3-28 



4 -7775 



9 -6181 





13 22 26 „ 





Moncalieri . 



+ 62 55 42 





4 "4755 



9 8340 





13 45 28 „ 





II. " On the Limits to the Order and Degree of the Fun- 

 damental Invariants of Binary Quantics." By J. J. 

 Sylvester, M.A., LL.D., F.R.S., Professor in the Johns 

 Hopkins University, Baltimore, U.S. Received December 

 26, 1877. 



The developments which I have recently given to Professor Cayley's 

 second method of dealing with invariants (the first method being that 

 which has been exclusively used by Professor Gordan), has led me 

 through the theory of the Canonical Generating Fraction to the 

 following results, showing that the degree and order of the funda- 

 mental invariants and covariants to a quantic or system of quantics 

 are subject to algebraical limits of a very simple kind, and I think it 

 right that these results should not be withheld from the knowlege of 

 those who are pursuing another and, as it seems to me, much more 

 arduous and less promising direction of inquiry into the same subject. 



By order I mean the dimensions of a derived form in the coeffi- 

 cients of its primitive (Clebsch and Gordan's grad), and by degree the 

 dimensions in the variables (Clebsch and Gordan's ordnung). 



First as to degree. 



If there be a system of n, n n" . . odd degreed quantics and v, . . 

 &c, even ones, then (with the exception of the case when the system 

 reduces to a single linear function or a single quadratic) the degree of 



