288 H. A. Newton on the height of Auroral Arches, 
from the earth, R the earth’s radius, } the distance from the ob- 
server to the point on the earth directly under that part of the 
cloud which forms the vertex of the arch, and c the distance in 
like manner from the observer to the point underneath that part 
of the arch seen in the horizon. We have then these equations, 
R-+2=Rsece=Reoshsec(b+h); . . .. (I) 
and since d, d—b, and c, are the three sides of a spherical tri- 
angle, and a the angle, 
cos(d—b)=cosd cose-+sindsinecosa. . . « « (2) 
From these equations b and ¢ may be eliminated, and x found 
in terms of a, d, h, and R. 
From (1) cos(b-+-h)=cosk cose, and hence 
cos (d —b)=cos }d-h—(b+h)? : 
008 (1-1) cos (+h) + sin (dA) sin (bh) 
=cos (d-+-h) cosh cose-+ sin (d-+-h)(1 —cos?h cose)”, (8) 
Equating (2) and (3) and dividing by cose, ; 
cos d-L-sin d tan c cos a=cos(d--h) cos h-sin(d-+-h)(sec2e—cos?h)*. (4) 
But cos d=cos(d+h)cos h+sin(d+A)sin h. Substituting, divid- — 
ing by sin(d+A) and placing sing for sind cosa cosec (d+h) we 
have 
sin h--tan ¢ sin p=(sec?¢ — cos? y3, 
Reducing, tanc=2sin h sin sec?9. 
Hence, to compute the altitude of the auroral arch above the 
earth’s surface, we have the three equations, 
sin p=sin d cosa cosec (d-+-h), 
tanc = 2sinh sin sec”, 
and r= R(sece—1). 
To apply these observations to particular cases I have wee 
25 or 30 auroral arches, observed by Pres. Stiles, Prof. pre 
Mr. Herrick, and Mr. Bradley. They were all observed at 4 
Haven, except those in the year 1860, which were seen by ma 
Bradley at Chicago. Most of these observations are pee 
Auroral Registers of Mr. Herrick and Mr. Bradley. These }?3" 
isters will form part of a velume of memoirs about to . at 
lished by the Conn. Acad. of Arts and Sciences. The va en 
d is assumed to be 82°, which is also very nearly the eee 
from New Haven to the magnetic pole of the earth. In the 
lowing table are given the dates of the auroras, the : oak 
apparent altitudes and amplitudes of the arches, and the 
puted values of 2 in miles, and in kilometers. __ [have 
In selecting from the Registers the arches for this yarns ro 
omitted those which were low in the north, as the horizo 
tent is then concealed by the mists. I have also om! ; 
of which the observations were indefinite, or seemed imperfect. ; 
pub: 
