138 Mr. J. J. Sylvester on the new Rule for finding Superior and 



periments of Chantrey and Blagdcn are often referred to as 

 illustrations of the surprisingly high temperature to which the 

 human body may for a short time be exposed without injury. 

 These experimenters owed their safety to two things, — to the 

 non-conductibility of their tissues, and the non-conductibility of 

 the air in contact with them. Were either of these materials 

 changed, the experiments could not have been made. If air 

 were a good conductor, and parted with its heat readily, their 

 hands and faces would have shared the fate of the beefsteak 

 and eggs which were cooked in contact with tin in the same 

 oven. Were their bodies good conductors, they would have 

 become heated like the tin, the heat would have been trans- 

 ferred to the deeper tissues and organs, to the probable destruc- 

 tion of the latter. As it was, however, both the causes mentioned 

 contributed to the success of the experiment, and a mere surface 

 irritation was the only inconvenience felt. 



XIX. On the new Rule for finding Superior and Inferior Limits 

 to the real Roots of any Algebraical Equation.' By J. J. Syl- 

 vester, F.R.S.* 



THE lemma accessory to the demonstration of the rule for 

 finding limits to the roots of an equation, given in the 

 addendum to my paper in the Magazine for this month, admits 

 of two successive and large steps of generalization, in which the 

 scope of the principal theorem will participate in an equal degree. 

 1. Whatever the signs may be of q v q 2 , q 3 , . . . q r , the deno- 

 minator of the continued fraction 



i i_ % j_ 



?1+ 9.2+ ?3 " 'Qr 



will have the same sign as q^ . q 2 . q 3 . . . q r , provided that 

 [ 5 V_ 1 ]>/V_ 1 + - — [y r ]> 



Mr- 2 rJ fir- 1 



where fi v fit, • . . fi r -i signify any positive quantities whatso- 

 ever; in the particular case where fi x = fi 2 — p 3 = . . . =fj, r _ 1 = l, 

 we fall back upon the lemma as originally stated. 



But 2nd. The lemma admits of another modification, which 

 will in general impose far less stringent limits upon the arith- 

 metical values of the series of ^s. 



Let all the possible sequences of q's be taken which present 



* Communicated by the Author. 



