648 REPORT— 1900, 



and (3) becomes 



{X(a- + kay) + ly- } (/S'' + pfi'-y + o-/3y-) + -Bry '(a" + A:ay) + py' = . (4) 



Equation (4) shows that the two foregoing singularities can coexist on a quintic. 

 Moreover, there is nothing in the preceding equations of condition which prevents 

 values being assigned to the constants which make the singularities at A and B 

 real. Hence a quintic can have fifteen real points of inflexion. If, however, a 

 quintic could have seventeen real points of inflexion, it would be possible to 

 determine the constants so that C should be a real crunode. This would require 

 that 



jy = 0, Zo- = 0, -srZ; = 



in which case the quintic would break up into factors. Hence, fifteen is the 

 maximum number of real points of inflexion that a quintic can have. 



The fact that in the case of cubics, quartics, and quintics, only one third of the 

 total number of points of inflexion can be real, leaves very little room to doubt 

 that this law is universally true in the case of all algebraic curves. 



11. On a Central-difference Interpolation Formula. 

 By Professor J. D. Everett, F.R.S. 



The best-known formulae for interpolation by ' central differences' are difficult 

 to carry in the memory on account of their unsystt-matic aspect, one law being 

 applicable to the odd and another to the even terms. This disadvantage does not 

 attach to the formula here proposed. 



Let M^) and m, be two tabulated values between which a value is to be inserted 

 at distance p from z/^ and q from u^, so that jo + 9 = 1 . 



If we regard u„ and u^ as ordmates joined by a straight hne cut in the 

 required ratio, the ordinate of the point of section is qii^+pUi, which is a first 

 approximation, exact when second differences vanish. A second approximation, 

 exact when fourth differences vanish, is 



where in conformity with the notation explained in my paper of last year,^ 

 Afi^<(, ASt<i are the central differences of the second order con-esponding respectively 

 to u„ «j. This is a sufficient approximation when g\ of the sum of the two central 

 fourth differences is negligible. The complete formula is 



^ (q + r) ■ . . (q-r) r r ^ {p + r) . . . (j >-r) »y 

 ^ 1 (2.+ 1) ^ ^ "« ^ ^ 1 (2/-+r) ^ "" 



where the numerators and denominators are factorials, all having 2?- + I factors 

 with unity as the common diflference. 



The only novelty about the formula is the simplicity of its form. Each pair of 

 terms is equivalent to a pair of terms in the second of the two ceutral-dift'erence 

 formulse of Stirling (Methodus Bifercntialis, Prop. 20), which is reproduced in 

 some modern works ; but that formula contains odd as well as even differences. 



Since ^-l is —p, and^> — 1 is -q, the above formula may be written 



<7Wo +FUi -'^ { (? + l)^S?/„ + {p + 1)a8m, } 

 



+P9<J> + ])(l±}) {(y + 2) AWu, + {p + 2)AWu, } + Sec. 



which (in the absence of a table of numerical values of the coefficients) is perhaps 



See p. 645 in 1899 Report. The full paper is in the Quarterlt/ Journal of 

 Mathematics No. 124, 1900. 



