38 
VARIOUS NOTES. 
[714 
A Trigonometrical Identity. 
cos (b — c) cos (b + c + d) + cos a cos {a + d) 
= cos (c — a) cos (c + a + d) + cos b cos (b + d) 
= cos (a - b) cos (a + b + d) + cos c cos (c + d) 
= cos a cos (a + d) + cos b cos (b + cl) + cos c cos (c + d) — cos d. 
Extract from a Letter. 
“I wish to construct a correspondence such as 
(x + iy) s + (x + iy) = X + iY, 
or, say, for greater convenience 
4 (x iy) 3 — 3 (x + iy) = X + iY; 
viz. if 
then 
x +iy — cos u, 
X + iY= cos 3 u. 
Suppose 3« 0 is a value of 3u corresponding to a given value of X + i Y, then the 
three values of x + iy are of course cosw 0 , cos f u 0 ± ; but I am afraid that the cal 
culation of u 0 , even with cosh and sinh tables, would be very laborious. Writing 
X + iY = JR, (cos © + i sin ©), 
the intervals for © might be 5°, 10° or even 15°, those of R, say 01 from 0 to 2, 
and then 0'5 up to 4 or 5; and 2 places of decimals would be quite sufficient; but 
even this would probably involve a great mass of calculation. 
It has occurred to me that perhaps a geometrical solution might be found for 
the equation X + iY = cos 3u.” 
October 31, 1877.