210 Mr. J. J. Sylvester on the New Rule of Limits. 



The phenomena of the rock veins in the masses of granite are 

 rather complicated. 



These veins appear generally to have formed at the time of 

 crystallization of the granite ; their great richness in quartz 

 favours this opinion, it being the last mineral of the rock to 

 remain in a fluid state. 



The granite of the Vosges forms smaller eminences around 

 the bosses of the Ballons, and is itself covered by stratified rocks, 

 into which it graduates by insensible degrees. The granite of 

 the Ballons has evidently penetrated with violence into the gra- 

 nite of the Vosges ; this is well seen at Mehachamp. In some 

 places the junction of the two is not discoverable. 



Of these rocks, then, that containing the smaller proportion 

 of silica and the greater of alumina is the more recent. 



The distinction of two granites in the chain of the Vosges is 

 not of mere local interest ; the remark may be extended to most 

 granitic regions, of which I will only mention the right bank of 

 the Rhine, Normandy, Brittany, Auvergne, Ireland, &c. 



The general application of the above observations shows that 

 the same geological phenomena are reproduced after long inter- 

 vals of time and in widely separated districts. It is not then 

 surprising that we should find in most granitic regions two gra- 

 nites : the one porphyritic, and containing but one mica ; the 

 other granular (grenu), and containing two micas; the former 

 being the more recent, and generally poorer in silica. 



XXXII. Note on the New Rule of Limits. 

 By J. J. Sylvester, F.R.S.* 



IT may appear like harping too long on the same string to 

 add any further remarks on the rule relating to so simple 

 and elementary a matter as that of assigning limits to the roots 

 of a given algebraical equation ; but it will be remembered that 

 some of the greatest masters of analysis, including the honoured 

 names of Newton and Cauchy, have not disdained to treat, and 

 to give to the world their comparatively imperfect results on this 

 very subject. I hope, therefore, to stand excused of any undue 

 egotism in adding some observations which may tend to present, 

 under a clearer aspect and more finished form, the new and beau- 

 tifully flexible rule laid before the readers of this Magazine in 

 the two preceding Numbers. 



1st. 1 observe that any succession of signs maybe considered 

 as made up of, and decomposable into, sequences of changes 

 exclusively, if we agree to consider, where necessary, a single iso- 



* Communicated by the Author. 



