340 



EEPOET — 1880. 



an even number of ordinates has an error numerically about double tbat 

 of tlie corresponding rule for one ordinate less. The number of cases 

 tried is not sufficient to warrant any general inference as to the compara- 

 tive amount of error, especially when we consider their signs ; but it is 

 highly probable that the rule with an odd number of oi'dinates is always 

 better arithmetically than, and not only of the same order of error as, 

 the rule with one- more ordinate. As has already been stated, no 

 general investigation of these comparative values appears to have been 

 made. The point is, however, one rather of analytical curiosity than 

 of real importance. The rules requiring high orders of differences are 

 better replaced by lower rules with more ordinates, unless in the very rare 

 cases where the ordinates themselves are difficult to calculate. It is 

 claimed that such an exception is found in calculating the curve of stability 

 of a ship when the mainwale, or armour shelf, and the deck are successively 

 immersed ; but there is at least a doubt in these cases whether the dis- 

 continuity, which makes the calculation of more ordinates difficult, does 

 not vitiate the accuracy of the higher ordei's of differences. If that be 

 so, the advantage sought by their use — namely, to be sure of not adding 

 an error of calculation to the errors of measurement, or to the errors due. 

 to wide intervals — would of course be lost. Nevertheless the higher rules 

 are analytically nearer the truth, and must be actually so in certain cases. 

 Only it must not be taken for granted that these are usual cases. It is 

 the practice of French naval architects to use the polygonal rule through- 

 out their calculations, in deliberate preference to the rule of three ordinates. 

 The arithmetical work is thereby much simplified, and so the liability to 

 accidental error is diminished. Moreover by taking ordinates sufficiently 

 close, the error of the rule can be reduced without limit, and where the 

 ordinates are inexact, it is not clear that the parabolic rule has any 

 advantage.* 



In dealing with actual data, the use of a large number of ordinates has 

 evidently the advantage of taking a more comjjlete account of the facts 

 than the use of a smaller number. Any want of continuity between the 

 ordinates is necessarily ignored by all the rules, and that to the greater 

 extent, the greater the interval. 



The rule of nine ordinates, and many of the higher rules, involve nega- 

 tive as well as positive coefficients, and are inconvenient on that account. 



The amount of the difference between the use of the polygonal rule 

 and the parabolic (or Simpson's first) rule is best shown geometrically 

 as follows : 



* Dr. Fan has used the same rule, or an arithmetical process equivalent to it, 

 for the integrations used in the Life Tables calculated under his superintendence by 

 the Eeoistrar-General's Department. See the Sixth, Eleventh, and Tivclfth lieports 

 of the Mi-yUtrar- General for Births, Deaths, and Marriages in England (1847, 1852, 

 1853), and the English Life Table published by the Kegistrar-General. 



