442 Mr. Henry Bowman on a Double Rainbow. 
Zgasin®2a H+R, 9.) 
OrRk -QeR 17 @ 
@csin®a K+R 
bina eatig igler* Sala REM ibe Sars Phaldl Gb beens 
In extracting these roots to obtain p and g, they must be 
taken with the same or different signs to fulfil equation (2.) ; 
and as they may under this restriction have two values, there 
are two different linear functions, and hence two points de- 
terminable, which will fulfil the conditions. 
Again, as in the solution already referred to, # and & can 
be found from (4, 5, 6.); and the general forms being the 
same, the values will be real, as they were proved to be in 
that place. 
Lastly, the value of r may be found from either of the 
equations (4, 5, 6.) ; and as it is of the first degree in those, it 
is evidently real. However, to obtain a symmetrical form it 
must be actually obtained from (6.), or from some symmetri- 
cal combinations of (4, 5.). 
Royal Military Academy, April 7, 1843. 
p=1—aa=1l—- 
. (10.) 
LXXVI. On a double Rainbow. By Henry Bowman, Esq. 
To the Editors of the Philosophical Magazine and Journal. 
OX Tuesday, March 21st, occurred here, between half-past 
2 o'clock and a quarter past 3 in the afternoon, a thun- 
der-storm, which, though by no means severe, continued for 
a considerable time. ‘The lightning was distant upwards of 
a mile, and the thunder was heard in short detached peals of 
nearly uniform duration, and almost at regular intervals, and 
had so little the character of ordinary thunder, that some time 
elapsed ere I recognized it to be such. It resembled more 
the alternate opening and closing of a window in an adjoining 
room. 
But the most remarkable pheenomenon attending this storm 
was a double rainbow, which appeared during and after a 
smart thunder-shower about 3 o’clock. The primary bow, 
which appeared some time before the secondary, was rather 
vivid, but the latter soon vied with it, until the two appeared 
with nearly equal brillancy. 
The arcs seemed to reach the horizon at both extremities, 
and to occupy perhaps 120° of a circle. Within the primary 
bow, I counted two or three reflexions or mppleniakasy bows, 
consisting chiefly of red and green rays. he bows were seen 
against a dense mass of cloud, of a fine neutral tint, and what 
