of Edinburgh, Session 1876-77. 347 
The quinquisections were obtained by help of the equation 
16 sin a 5 - 20 sin a 3 + 5 sin a - sin 5a = 0 = E . 
Regarding E as a function of sin a, its successive derivatives 
are 
X E = 80. sin a 4 - 60. sin a 2 + 5 
a E = 320. sin a 3 -120. sin a 
3 E = 960. sin a 2 — 120 
4 E = 1920. sin a 
5 E = 1920 
now, while resolving the equation, we get the values of all the 
derivatives ; so, taking advantage of these, we have 
cos a 2 = ^ — -i- ..E 
8 960 3 
cos 2a = ? - 4— 3 E 
4 480 3 
sin 3a - ~ sin a - 2 E 
2 80 
cos 4a = — 
1 
4 480 
E 4 - — E • 
3 + 10 1 ’ 
the first of these gives us, by extracting the square root, cos a. 
Applying this method to the arc l c 25', the eightieth part of the 
quadrant, I obtained the sine and cosine of 25', and proceeded to 
compute the functions of its multiples by help of second differences, 
according to the well-known formula. 
sin (w + 1) a - 2 sin na + sin (a— 1) a = sin n a. 2 vera; 
and, because the multiplier 2 vera is to be repeatedly used, a table 
of its multiples was constructed, in the case of a = 25', up to the 
hundredth, in the cases of a =5' and a = 1', up to the thousandth 
multiple. 
The sines of these multiples being computed continuously, an 
error in any part of the work, propagated subsequent errors, which 
could not possibly be overlooked in comparing the results at each 
