Ml". P. Nicholson o>i derivative Analysis. 433 



It is presumed that what has been stated is quite sufficient 

 to show that the objection is altogether groundless. And now 

 to conclude, with a short recapitulation of" the merits of the 

 question : It is to be noted, on the one hand, that in the sus- 

 pension bridge built on the principle of the catenary curve, 

 the weight of the platform, chains, &c. (without any loading), 

 will throw a strain of 7^ tons per square inch on the suspend- 

 ing chain ; and that this strain may with great probability be 

 increased to 1 5 tons : while, on the other hand, by employing 

 the diagonal rods, the weight of the platform would produce 

 a strain of only 3^ tons per square inch, which strain could 

 never exceed 7 tons per inch. In short, the strain in the 

 new plan would, under a parity of circumstances, be always 

 less than half what it would be in the old plan. 



XC. Derivative Analysis ; being a new aiid more comprehensive 

 Method of the Transformation of Functions than any hitherto 

 discovered : extending not only to the Extraction of the Roots 

 of Equations, but also to the Reduction of Qiiantities from 

 the Midtiples of Povoers or Products to other equivalent Ex- 

 pressions, by xi'hich the Summation of any rational Series may 

 be readily effected. By Mr. Peter Nicholson. 

 5 Claremont-place, Judd-street. 



[Concluded from p. 355.] 



TRANSFORM the function Sj'' + 4x— 1 into the function 

 3(.r — 0-2)' + B (a- — ()-2)-|-C2or tofindan equivalent func- 

 tion in V, where v shall be two tenths of unity less than x, 

 that is x = v + (i-2 



— 0-2 i 



— 0-2 i 



4.-0 -1 



3x-2 = -6 4-6 X -2 = -92 I —-08 

 6 5-2 



We may here observe, by the by, that since to extract the 

 root of an etjuation is nothing more than to diminish that 

 root by the whole of itself, that is, by taking away the whole 

 of tlie absohite number, we have now taken away -2 from the 

 root of the (juadratic eciuation 3.r' + 4.r— 1 =0; and if we con- 

 tinue the same process by taking away a part, we shall at last 

 arrive at the root, or as many true figures of the root as are 

 found. In order to find a secontl fimn-e in the root, we have 

 only to annex a cipher to the absolute number -08, and di- 

 vide it by the preceding coefficient 52 ; then the number of 



Vol. G2. No. 308. Dec. 1823. 3 I integers 



