ON QUADEATDEES AND INTEBPOLATION. 331 



.-. 1 Cdx = 1 : log (1 + A) = - 1 : log (1 - F) 



= F-> - Vi + VoF - V3F2 + 



= 1 + A-' - Vi + VoF - VsF'^ + 



Now introduce the limits of integration, using ascending diflferences at 

 one end and descending at the other. This gives 



I A' ycU = y„ + A-' 7/,, - Vi y„ + Vo F 2/„ 



-(A-iyo + V,7/o + V2A2/0 + . . . .) 



= ¥2/0 + 2/1 +2/2 + + 2/2-1 + \yn 



- V. (A 2/0 - F2/„) - V3 (A2 2/0 + F2 y„) 



= V,+, I A'yo + (-)'■ F,//.}- 



or, since F^ y„ = A' (1 + A)"'' . y,, = A-" y„_,., 

 we have, finally, 



h f x'y^^ "^ ^ ^^° "^^' + 2/2 + + 2/<i-i + i2/« 



- V2 (A2/0 - A7/,_0 - V3 (A^^o + A2 y„_o) - 



- Vh-1 I a,, 1/0 + (-)'"^''2/-i->- j 



which is the usual formula of quadratures.* 



Section 2. — Inverse Tntei'jwlation. 



Using the common formula of direct interpolation, the problem is to 

 find the value of n from the formula 



V;, = 00 + « A ./-o + "—^ A^ 00 + 



where everything but n is known. 



Stopping at any given order (say the r*) of differences, neglecting all 

 beyond it, which is evidently permissible provided the difiei-ences be con- 

 vergent, and sufficiently so (and only on that hypothesis) this really 

 involves the solution of an equation of the r*'^ degree in n. It is usual 

 to efiect the solution by successive approximation, that is to say, by 

 stopping successively at the first, second, third, &c., difierences in succes- 

 sion, and determining at each step a new and more accurate value of n. 



Thus, neglecting all beyond the first difference, a first approximation 

 gives 



This merely amounts to the use of proportionate parts. 

 The second step is to calculate 



i(jl, - 1) A2 0o = Ci, 



* See De Morgan, Biff, and Int. Calc. p. 313. The proof there given is sub- 

 stantially the same as this, only differing in arrangement. The above arrangement 

 perhaps shows a little more obviously the reason why the constant part used is the 

 same in both formuliB. Stated more exactly, what is done is to introduce an inter- 

 mediate (but indefinite) limit, and to reverse one of the definite integrations. 



