But taking, for example, Simpson's mean values at Karasjok, we have observed p - I + - I i = 0.05 X (cm.
This involves the use of Cartesian co-ordinates, and leads to important general formulae, such as Simpson's formula.
The application of Simpson's rule, for instance, to a plane figure implies certain assumptions as to the nature of the bounding curve.
For this reason, formulae which will only give approximate results are usually classed together as rules, whether the inaccuracy lies (as in the case of Huygens's rule) in the formula itself, or (as in the case of Simpson's rule) in its application to the data.
This is Simpson's formula.
This is Simpson's second formula.
Simpson's two formulae also apply if u is of the form px 3 - }- 5x 2 + rx -}- s.
Simpson's (first) formula, for instance, holds for f = I, and is obtained by taking p = i and ignoring differences after 52up.
Thus we find from (i) that Simpson's second formula, for the case where the top is a parabola (with axis, as before, at right angles to the base) and there are three strips of breadth h, may be replaced by area = 8h(3u i + 2U 1 + 3us).
This might have been deduced directly from Simpson's first formula, by a series of eliminations.
It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid.
The areas of the sides for which 0 and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second formula; call these Ao and A2, and Al.
Hence Simpson's formula applies, and the volume is lh(A0-I-4A1+ A2).
Applying Simpson's formula to each of these, we have A -__9= *h(uo + 4 14 1 + u2) + 3 h (14 2 + 4u3 + u4) + ..
+ 214m-2 + 4 u m-i + um) This is Simpson's rule.
(ii) Similarly, if m is a multiple of 3, the repeated application of Simpson's second formula gives Simpson's second rule A 1?
Consider, for example, Simpson's rule (§ 68 (i)).
+ u m-p + zum), which may be denoted by Cp. With this notation, the area as given by Simpson's rule may be written in the form sC l - 3 C2 or CI+ 1 3 `-(C1C2).
(i) m a multiple of 2 (Simpson's rule).
(ii) m a multiple of 3 (Simpson's second rule).
The formula applied can then be either Simpson's rule or a rule based on Gauss's theorem for two ordinates (§ 56).
These theorems may prove useful in preliminary calculations where the pressure-curve is nearly straight; but, in the absence of any observable law, the area of the pressure-curve must be read off by a planimeter, or calculated by Simpson's rule, as an indicator diagram.
Besides these public establishments for the custody of lunatics, there are in the vicinity of Dublin various private asylums. The principal institution for blind men (and also those afflicted by gout) is Simpson's hospital (1780), founded by a merchant of Dublin; while blind women are maintained at the Molyneux asylum (1815).
An admirable biography is to be found in Richard Simpson's Edmund Campion (1867); and a complete list of his works in De Backer's Bibliotheque de la compagnie de Jesus.