202 Transactions.- — Miscellaneous. 



Art. IX. — On the Construction of a Table of Nakiral Sines by 

 Means of a Nciv Relation between the Leading Differences. 



By C. E. 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.,* and are : — 



Sin 9° = 015643, 44650, 40230, 86901. 0105 

 Cos 9° = 0-98768, 83405, 95137, 72619, 0040 



To test these values they are squared, and give : — 

 Sin 2 9° = 0-02447, 17418, 52423, 21394, 1780 

 Cos 2 9° := 0-97552, 82581, 47576, 78605, 8219 

 hence the values are correct. 



Now, k = 2 (1 - cos Ax) = 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 = Ax = 9° 



sin 0° = 0-00000, 00000, 00000, 00000, 0000 



A sin 0° = + 0-15643, 44650, 10230, 86901, 0105 



sin 9° sir 0° + A sin 0° = + 0-15643, l 1650, 40230, 86901, 0105 



A2 sin 0° = - k . sin 9 = - 0-00385, 19357, 055 1 I. 31391, 79172 



A sin 9° = A sin 0° -+- A- sin 0° = + 0-15258, 25293, 3 1716, 55509, 218S 



A sin ii = - k . A ei = - 0-00375, 70882, 64602, 87371, 61559 



A- sin 9 C A' sin 0° + A 3 sin0° = - 0-00760, 90239, 70117, 18763, 40731 



a 4 sin 0° = - k . a 2 sin 9° = + 0-00018, 73594, 23047, 03150, 04117, 



13 

 A 8 Bin 9° - A :; sin 0° + A' 1 sin 0° == - 0-00356, 97288, 41555, 84221, 57442 



A* sin 0° = — k . A 8 sin 9° - + 0-00008, 78985, 71329, 89818, 8990G, 



78 

 A« sin 9° = A* sin 0° + A 5 sin 0° = + 0-00027, 52579, 94376, 92968, 94023, 



A' ; sin 0° = - k . A' 1 sin 9° = - 0-00000, 07777, 65350, 46850, 65814, 



220 



* Trans. N.Z. Inst., 1902, p. 409-10. 



