C—Ja

Prediction

The predictions have been carried out generally in a similar manner to thai of last year. The following modifications have, however, been made. The M curve is now composed of all the M tides (Mi, M,. M... M 4, and Mg); the 8 curve of S t , S 2 , S 4 , and S 6 tides; and the X curve of X, and Iv., tides. In Figs. 1 and 2 the separate X, and X., curves are shown as well as the combined curve X, + K 2 , both for Auckland and Wellington, and on the same scale. The assumption involved in the use. of the template curves is that the resultant curve, made up of the M curve with the addition of the collections due to the other tides, is practically an M a curve of different amplitude. That this assumption is justified within practical limits is shown by the tests of the predictions, which agree very closely with actuality. As a further check on this assumption. whenever the correction to the M s tide of 11. or L. water approaches I hour, a fourth measurement is made of the combined curves 2 hours from the time of M 3, 11. or L. water, and it has been found that the template curve invariably goes through the fourth point, so determined. An example of this check is shown in Fig. 3, where \('KJ represents the M combined curva For the looth L.W. of Auckland, 1913. AB, CD, EF are the three measured corrections due to the other short-period tides. The template curve (&=-8) is fitted over the points BDF, keeping its centre-line Fll parallel to CD, EF, and AB. In this case the apex to the curve coincides with K. and the correction to the time is —1 h., while the height is TB3 ft. The fourth measurement made at —2 h. gave the correction JI, and it was found that the template curve passed e.xaclK through ihe point 1, thus showing, as far as the scale of the drawing permits, (hat the assumption is correct. ('heck of Predictions. Through the courtesy of the Engineer to the Auckland Harbour Board (Air. W. 11. Hi ir M.lnst.C.E.) the Auckland tide-records for 1912, January, were sent here as soon as they wenavailable, and a portion of the check fo the prediction is shown in Fig. I. where the refills for the January 25th to 28th are shown plotted on a copy of the tide-record. The whole of the January record was checked in this way, and gays equally good results. Datum Levels. As in the case of Wellington, the suggested adoption of the datum of the Indian spring lowwater mark for Auckland would result in the datum being— Tide. Hemi-rangc. M 2 .. .. .. .. 3-814 11. S 2 .. .. .. .. 0-583 ft. XtK t .. .. .. .. .. 0-233 ft. 0 .. .. .. .. .. 0-059 ft. 4-689 ft. below mean sea-level. In this case also the comparatively large tides— Tide. Semi-range. N 2 .. .. .. .. .. 0-797 ft. v .. .. .. .. .. 0-236 ft. L .. .. .. .. 0-221 ft. MS . . .. . . . . . . 0-169 ft. K 2 .. .. .. .. 0-145 ft. /t .. .. .. .. .. 0-126 ft. P.. .. .. .. .. 0-068 ft. are all greater than the O tide. Other Methods of Harmonic Analysis. Following a valuable suggestion made by Dr. P. H. Cowell, .Superintendent of the Nautical Almanac. London, a method of analysis of tidal observations, where the observations are summed first every 24 mean solar hours, and secondly every 25 mean solar hours —has been partly developed. An example as applied to the M a tide will show the general procedure, and reference should also be made to Dr. Borgen's papers. For this and other tides which have a period nearer to 25 than to 24 hours, the observations are summed every 25 hours throughout the year: thus, for the Auckland tides the following schedule of sums is made : — List of Sums, beginning 1908, December I. Day. Oh. 1 h. 2 h. 22 h. 23 h. 21 h. 0 80 85 78 II 59 71 1 160 167 151 .. 88 117 112 2 239 249 230 .. 130 176 214 353 17891 17574 17220 .. 18363 18281 18087 354 17920 17600 17251 .. 18418 18325 18121 In this schedule the hours arc mean solar, the day is an arbitrary one of 25 hours, and the unit for heights is J ft. The height of the tide above its mean level: at any hour t (counted from oh to 21 h), on any day i' (beginning with o), may be represented by a sum of terms of the form— hi, v= R cos (i t-C+v.2s i) + B. cos _£, + ~. 25 i y ) + (1) where R = H/ r= X - 'V'+M).

5—G. la.

33