On the Notations of the Calculus. 25 



1 <2 i • • • 



b v b, b , . . . 



Let X and X^ , , be any two consecutive limiting poly- 

 nomials ; then, since X and X , , have like and contrary 



signs immediately before and after the passage of a root of 

 the equation X =0 (Young on Equations, art. 76), it is 



manifest, by inspecting the above arrangement of the roots, 

 that one variation, and only one, will be introduced on the 

 passage of each root of the equation X = ; the value of x 

 being supposed to continually decrease from the greatest root 

 downwards. Now, since all the positive roots are comprised 

 between and oo , it follows, from what is proved above, that 

 the number of variations arising from making * = will ex- 

 hibit the number of positive roots in the equation ; which va- 

 riations, it is manifest, are the same, both in number and order, 

 as those of the original equation. 



It is proved in exactly the same manner as above, the value 

 of x being supposed to increase from the least root upwards, 

 that no equation can have a greater number of negative roots 

 than permanencies, or successive repetitions of the same 



si £ n * 



Cor. — It is also plain that, if any two numbers be substi- 

 tuted for x in the functions X, X v X 2 , X 3 , &c, the difference 

 between the number of variations, in the signs of the results 

 of these substitutions, will express exactly the number of roots 

 comprised between these two numbers. 



Yours, &c, 

 Preston, November 5, 1843. SEPTIMUS Tebay. 



VI. Observations on the Notations employed in the Differential 



and Integral Calculus. By J. J.* 

 HPHE differential and integral calculus are applied to nearly 

 "*• the whole circle of the physical sciences ; scarcely any 

 treatise on mechanics, optics, astronomy, &c. can be read so 

 as to be understood without a thorough knowledge of these 

 extensively useful adjuncts, or at all events without a pretty 

 close acquaintance with them. It is clearly expedient then 

 that sciences so generally applied and so constantly occurring 

 should be kept as simple as possible. The symbols employed 

 should be as free as they can be from ambiguity, at the same 

 * Communicated by the Author. 



