92 Calculation of the horizontal Refraction, Sc. 
scent. The formula will therefore be reduced to the identical 
equation g = q. 
By substituting the values of p and q, and then dividing by 
A, we get 
3 
Bate 
x 
—% 2 
UC Bt AC - w+ —.03 + &e.; 
wherefore 
So —32 
—dX 220 32A 2c 
du=dxx $c +- —.x2 + 2? + &e. 2, 
U 1 1.2 5 
and hence 
C—2r ——3X 
de te: aa 24 322% 
aie: Sx} Trg eer a Comes 
2i 1 . 
t feet 
v 
All the terms of this expression are now integrable. For if we 
1 : : 
put 1h jest ae t, then i = —: le; and, taking the inte- 
m f 
grals between « =0, ¢=0 and x=1, t =1, we have 
x denoting the semicircumference of which unit is the radius. 
' Thus we get 
Bax ee aahn  Mipie Releclaee 
ees ee eee ee) 
af 2i 1 afi 1.2 Wf 3 
and the series may be continued ad libitum, since the law of con- 
tinuation is known, 
If we expand the exponentials we shall obtain, 
: f aN 32 9 22 aly 
a7 Oram V2 ) 
23 43 33 23 
+ 38 (T= 85S +385 -)) 
a4 5a 44 9A o4 
— —4—— — —4—_ + ] 
+ isalGe thing aii ) 
+ &c. 
Both these serieses were found by Kramp and Laplace* ; but 
these geometers proceeded in an inverse order to that followed 
* Ref. Ast. pp. 119 and 120; Mécan. Céleste, vol. iy. p. 252. 
here. 
