166 On the new Method for 
w; by which means the foregoing equation (3) will be converted 
into one containing only two variable quantities. 
In the ase of the horizontal refraction we have sin A = 0, 
cos A = 1; and, the series for w containing only the even powers 
of 7, it will be determined by the single equation, 
ddw _ i f Cine } 
dr2 pe r da . 
The calculation will be rendered more simple by putting r=e x 
ig 
>; for then, 
A i 
ddwa ds 
— —A 
d dw 
A ’ (4) 
r=e x —=- 
Apok 
and we have now to determine the series, 
w= Co*+ Egt+ Go + &e. 
In order to bring the question of the convergency to a deci- 
sion, the best way will be to examine the case of the horizontal 
refraction in a particular hypothesis; for instance, jn that of a 
uniform temperature prevailing in the atmosphere. In this hy- 
pothesis, the densities are proportional to the pressures ; that is 
- =x = 1—w. Wherefore the first of the equations (2) will 
dS eek 
become (l—w) = f’—ds(l—«); whence —- =>. The 
equation (4) will therefore become, 
ddw si ek e 2 
des —A== (IA) Pe Hath Se 
The coefficients of the series for » will be determined by sub- 
stitution, as usual, viz. 
1a 
ig ae 
_ lea 
age? 
1A 1 1—-A\2 
Gr sent CE) 
1230 * 30 Te te 
&c. 
Without carrying the calculation further, we may observe that 
the series will contain the part, 
Mi, 7s. Lif lea \? wane 4 
wie et + gee) + a(S) 8? + Be 
which, as g* is abont 2, will converge very slowly. We may 
therefore conclude with certainty, that the method of calculation 
proposed by Dr, Young, is deficient in convergency, that is, a few 
of 
” 
