Art. IX.—On the Construction of a Table of Natural Sines by Means of a New Relation between the Leading Differences. By C. H. Adams. [Read before the Wellington Philosophical Society, 2nd November, 1904.] Part II. 1. To show the immense power of this method of obtaining the leading differences, an exceptional example is here given of the formation of a table of natural sines to twenty-four decimal places for every nine degrees of the quadrant. The values of sin 9° and cos 9° are readily obtained from the series given in Part I.,* Trans. N.Z. Inst., 1902, p. 409–10. and are— Sin 9° = 0.15643, 44650, 40230, 86901, 0105 Cos 9° = 0.98768, 83405, 95137, 72619, 0040 To test these values they are squared, and give:— Sin2 9° = 0.02447, 17418, 52423, 21394, 1780 Cos2 9° = 0.97552, 82581, 47576, 78605, 8219 hence the values are correct. Now, k = 2 (1—cos Δx) = 2 (1—cos 9°) = 0.02462, 33188, 09724, 54761, 99195. The leading differences are formed as described in Part I, and a convenient working schedule is arranged thus:— Tabular interval = Δx = 9° sin 0° = 0 00000, 00000, 00000, 00000, 0000 Δ sin 0° = + 0.15643, 44650, 40230, 86901, 0105 sin 9° = sin 0° + Δ sin 0° = + 0.15643, 44650, 40230, 86901, 0105 Δ2 sin 0° = - k. sin 9° = - 0.00385, 19357, 05514, 31391, 79172 Δ sin 9° = Δ sin 0° + Δ2 sin 0° = + 0.15258, 23293, 34716, 55509, 2188 Δ3 sin 0° = - k. sin 9° = - 0.00375, 70882, 64602, 87371, 61559 Δ2 sin 9° = Δ2 sin 0° + Δ3 sin 0° = - 0.00760, 90239, 70117, 18763, 40731 Δ4 sin 0° = - k. Δ2 sin 9° = + 0.00018, 73594, 23047, 03150, 04117, 13 Δ3 sin 9° = Δ3 sin 0° + Δ4 sin 0° = - 0.00356, 97288, 41555, 84221, 57442 Δ5 sin 0° = - k. Δ3 sin 9° = + 0.00008, 78985, 71329, 89818, 89906, 78 Δ4 sin 9° = Δ4 sin 0° + Δ5 sin 0° = + 0.00027, 52579, 94376, 92968, 94023, 91 Δ6 sin 0° = - k. Δ4 sin 9° = - 0.00000, 67777, 65350, 46850, 65814, 220