in an Atmosphere of uniform Temperature. 91 
last, and r being the horizontal refraction sought, if we make 
A =0, cos A=1, sin A=0, in the second of the equations (2), 
p. 165, we shall get 
dw 
/ 2is—2eBw : ' 
In the hypothesis of a uniform temperature, the densities are 
r= 6 ™ 
proportional to the pressures, and we have 2 = | —w: where- 
fore the first of the equations (2) will become 1 — w = f'—ds 
dw 1 : 
(1—w); whence ds = -——; ands= 1—.. As the integral 
extends from w = 0 tow= 1, if we puta = 1—w, and A= 
—}; we shall have s = l_, and 
B x f° du 
eas ——————————— J 
Mv 2i av ioe 
uw 
— —A +AU 
T= 
the integral being taken between the limits «= 0 andw=1, In 
order to accomplish the integration, I assume 
1 1 
[— = 1--— A+ AU; 
or, la=lu+ta—Aaun; 
and, by taking the numbers corresponding to the logarithms, 
we get 
=A =i 
o*r=uc ", 
c being the number whose hyp. log. is unit. 
Let p=ac*x x, g=Au; then 
pager: 
and we must now find q ina series of the powers of p. This 
may be effected by expanding the exponential quantity and then 
reverting the series; or by other well known methods by means 
of which the law of the terms may be discovered. I have found 
42 
g=P + P+ TGP + Tag Pt + Be 
n=2 
the general term being, esa xp”. The truth of this 
formula will be proved by substituting qe 4 for p, and then ex- 
panding all the exponentials, For it will appear that every 
power of q is multiplied into a coefficient of this form, 
n” + m.(n—1)” + m.— (n —2)"++ &c. 
m being less than 7; an expression which is known to be evane- 
M 2 scent, 
