OF ARBITRARY CONSTANTS, &c 127 



VO616, 1*2208, 1-5116, 2-0053, &c., and the successive differences of the series of factors headed 

 by unity are 



A'= + 0-0616, A'= + 0-0976, A'= + 00340, A*= +0-0373, &c. 

 These differences when multiplied by 14 are so small that in the application of the 

 transformation of Art. 8, for which in the present case q = 1, the differences may be neglected, 

 and the series there given reduced to its first term. It is thus that the remainder given above 

 was calculated. 



The sum of the series is now to be reduced to the forp p (cos 9 + \/- l sin 6), and 

 thus multiplied by e~''"^ and by x~i. We have 



for series log. mod. = 0-0036832 amp. = + o* 33' 58". 21 



for exponential log. mod. = 1-7371779 amp. = + ISO* 49' o". 78 



for x'i log. mod. = 1-9247425 amp. = - 22* 30' 



1-6656036 + ]08« 52' 58". 99 



When the amplitude of x is 150", there are both superior and inferior terras in the ex- 

 pression of the function by means of descending series. It will be most convenient, as has 

 been explained, to put in succession, in the function multiplied by C in equation (20), 

 amp. x = 150" and amp. a? = - 210°, and to take the sum of the results. The first will give 

 the superior, the second the inferior term. 



For the amplitudes 90*, 150* of *, or more generally for any two amplitudes equidistant 

 from 120*, the amplitudes of xi will be equidistant from 180°, so that for any rational and 

 real function of a?^ we may pass from the result in the one case to the result in the other by 

 simply changing the sign of v — 1, or, which comes to the same, changing the sign of the 

 amplitude of the result. The series and the exponential are both such functions, and for the 

 factor x-i we have simply to replace the amplitude - 22* 30' by - 37° 30'. Hence we have 

 for the superior term 



log. mod. = 1-6656036 ; amp. = - 168* 52' 58". 99. 



When amp. x is changed from 150° to - 210°, amp. x^ is altered by 3 x 180°, and there- 

 fore the sign of x^ is changed. Hence the log. mod. of the exponential is less than it was by 

 2 X 1-737... or by more than 3. Hence 4 decimal places will be sufficient in calculating the 

 series, and 4-figure logarithms may be employed in the multiplications. The terms of the 

 series will be obtained from those already calculated by changing first the signs of the imagi- 

 nary parts, and secondly the sign of every second term, or, which comes to the same, by 

 changing the signs of the real parts in the terms of the orders 1, 3, 5..., and of the imagi- 

 nary parts in the terms of the orders 0, 2, 4... Hence we have 



+ 0-9914 + 0-0076 \/- 1 

 log. mod. = 1-9963 ; amp. = + 26'-5. 



