300 van orstrand: circular functions 



series. This computation is not a difficult task for the reason that 

 tables of the factorials 1 have been published, and a table of the 

 successive powers from 2 to 20 of the first 20 natural numbers 

 may be readily formed and checked by comparing the 20th differ- 

 ence with 1 20. The computation of terms of the form x n -4- n 

 is then easily made and checked by differences. The summation 

 of these terms need not be checked independently for the 10th 

 interpolation must agree with the value obtained by direct sub- 

 stitution in the series. 



If the interpolation formulas are written in the forms, 



Ax (Arc) 2 



sin (x + A x) = sin x + jj~ cos x — , sin x 



(Ax) 



3 



COS X + 



Ax . (Ax) 2 



cos (x + A x) = cos x — jy sin x — y~^~ cos x 



(Ax) 3 

 + I o sin x + . . . , 



it will be noted that the two series together contain terms of 

 the forms (Ax) n sin x -f- I n and (Ax) n cos x -f- \n where n assumes 

 successive values of the natural numbers beginning with unity. 

 We thus have two series of terms, 



1 

 sin a:, t^~ sin x, 



1 

 cos x T^p COS X, COS X, 



[ jj I 6 \n_ 



which may be evaluated by dividing the sine or cosine, as the 

 case may be, first by 2, this quotient by 3, the last by 4, and so on, 

 thus avoiding the use of one large factor. The only effect of the 

 factor (A:r) n is to shift the decimal point, and this may be deter- 

 mined once for all. The following details illustrate the method: 



1 J. W. L. Glaisher, Tables of the exponential function. Camb. Phil. Soc, 

 Trans. 13 : 246-247. 1883. 



