536 Transactions. — Chemistry and Physics. 



Enough has now been said to guard against misleading use 

 of the symbols + and — in graphical work. 



Notes on the Gkaphs. (Plates XXXVI., XXXVII.) 



Graph A gives the characteristic curve of deviations of 

 a cubic-formula curve to which a parabolic approximation 

 is applied (section 20) ; Graph B is that of section 21 ; 

 Graph C that of the " Mississippi Problem," Appendix, I. 

 Graph I. shows the curve of the standard parabolic and 

 Graph II. that of the cubic. Graph III. shows two 

 symmetrical quartics — (A), zero at x — 0, 1/3, 2/3, and 1, 

 and having the formula — 



(A) = 1x - lite 2 + 18a; 3 - 9x\ 



and (B), zero at x = 0, 1/4, 3/4, and 1, with formula — 



(B) = 3x - 19a; 2 + 32a; 3 - 16a; 4 . 



The maxima reach Oil and 025 respectively, or 1-2 and 

 1-6 per cent, of the numerical value of the coefficient of x i . 

 Graph IV. is a symmetrical quintic, zero at x = 0, 1/4, 1/2, 

 3/4, and 1, with formula — 



(C) = Sx - 25a; 2 + 70a; 3 - 80a; 4 + 32a; 5 . 



Its maxima reach 0-11, which, the coefficient of x h being 32, 

 represents - 34 per cent, of the value of the latter. Graphs 

 V., VI., and VII. are specimen standards of the hexic, heptic, 

 and octic degree respectively. Their formulas are given in the 

 accompanying table. They are all made zero at the sarnie 

 points as the previous quintic. The curves all being sym- 

 metrical, about x = 0*5, are, as has been before noted, both 

 simpler in form and more easy to compute values from if the 

 origin of the abscissa is taken at x = -J and the scale of the 

 variable halved. The formulae are accordingly given in terms 

 of z = 2x — 1, as well as in x — which is the most straight- 

 forward variable for common cases of a few terms. 



It will be remarked that, so far as these standard formulae 

 have been developed, it has been arranged to keep the formula 

 simple. In constructing formulae for actual practice, how- 

 ever, it may be better to sacrifice the mathematical simplicity 

 altogether in order to obtain curves that are convenient for 

 the visual processes of analysis (see Postcnpt). 



The percentage values of the maximum ordinates in these 

 curves compared with the value of the coefficient of the 

 highest power of x are as follows : Hexic, 039 per cent. ; 

 heptic, 03, 5 per cent. ; octic, 001, 3 per cent. Thus, if we 

 throw a formula into the standard form we can see what the 

 effect will be if we throw away the highest term. It is evi- 

 dent that we shall often be able thus to reduce a formula of 



