ON QUADRATURES AND INTERPOLATION. 369 



VII. — PEKIODICITr. 



The ordinaiy assumption of interpolation is that there shall be no 

 periodicity in the function, and this assumption is involved in the 

 apjDroximate equation virtually assigned to the carve being of parabolic 

 form. Any pei'iodicity vitiates the accuracy of the result, and the 

 detection of this periodicity is necessary before any correction can be 

 applied, or any special methods adapted to the jjeriodic character. 



In observed results, rather long series are required before periodicity 

 can be detected, unless it can be independently inferred from analytical 

 or phj^sical considerations. It is best detected by plotting a curve of 

 the function, and the process may be facilitated by first transforming 

 the function so as to deprive it of any very abrupt curvature. The 

 oscillation will then generally become visible, or may be made so by an 

 elliptic exaggeration of ordinates, taken nearly normal to the general 

 direction of the curve. 



The arithmetical methods of detecting periodicity are mere trans- 

 formations of this geometrical principle. They are very difficult and 

 intricate pieces of work, especially when the periodicity is of high order 

 and small amplitude. Examples may be seen in the discussions of the 

 inequalities of the planetary systems in astronomical works, and, in a less 

 elaborate way, in the discussion of the various periodicities which have 

 been associated with the sun-spot period.* They also joresent themselves 

 in the discussion of tides ; but in these cases the probability of a period is 

 sufficiently evident to cause it to be looked for in the proper way. 



"When the period is once found, there is seldom much difficulty in 

 dealing with it, either for interpolation or for quadrature. 



VIII. — Systematic Computation op Quadratures and Interpolations. 



In all work connected with either interpolation or quadrature it is 

 necessary, both, for convenience and correctness, to do the work in a 

 neat and well-arranged tabular form. The expression given in many 

 books for the parabolic quadrature, namely, ' to the sum of the first and 

 last ordinates add four times the sum of all the even ordinates, and twice 

 the sum of all the other ordinates, and multiply the total by one-third of 

 the intei'val ' is not the form in which any practised computer would 

 think of working. The slight repetition of labour involved in the tabular 

 form is as nothing compared with the trouble and chance of error involved 

 in disturbing the regular order of the ordinates. Moreover, when moments 

 ai'e required, as well as mere area, the tabular arrangement is a clear 

 gain of work. An example of the arrangement for obtaining the centre 

 of gravity of a curvilinear area is given below. The curve selected for 

 integration is y=2 \/ (x -|- 1) — 2, for the values 0, 1, 2, 3, 4, 5, 6 of a;. 



The result is that the area is 11'35G8, and that the coordinates of 

 the centre of gravity are 



43-55a,6 „.j,.-„ , 13-2840 , , ,.. 



* See also Messrs. C. & F. Chambers 'On the Mathematical Expression of Obser- 

 vations of Complex Periodical Phenomena,' Phil. Tmns. vol. 165 (for 1875") p 3(!1 

 1880. BB 



