8 J, LeConte on the Silver-Spring in Florida. 
Let . ii A =h, (the height of E above the water). 
=d, the depth of first fathom mark). 
3 TF’ FE’ Fete. =L, (the length of the fathoms). 
it #; a ,etc., — e angles of incidence. 
- " r', rl etc., =the corresponding angles of refraction. 
“ =the index of meee for water=1-336. 
Breti the law of refraction, sin r=n Xsin 4, and sinr’=nXsin 7, 
etc. By geometry, the angle iibtendad at E by FR’=i-?7: 
also the angle at HK’ after refraction =r—7’. Then by ap 
+2 
- nee az . go 
etry we have cotz hd) Xtan 6} cot 2 > (eedyx tans 
Hence it follows, that when A, d, L, and 7 are given, is pL elt. ate. 
ae be calculated, and conseque ntly, *,?", xe ete., may b 
found: hence the angle subtended at E by PE (= =i—1) as well 
as that subtended at H’ (=r—*’) by the same fathom-interval, 
ene known. For the sake of illustrating this is point, let us as- 
me h=2 feet; d=1 foot; L=6 feet; and 7=30°. Then, by 
the application of the formula above given ‘and the geometrical 
and physical principles already indicated, the following table 
has been calculated for the foregoing condition of things: 
Ratio off 
* 1h to EB’ 
| Angle su>tended at E. Angle subtended at E’ 
n| ¢ 
a # 41° 54/46" By F FY —19° 6/24" By F FB! —279 17! 186 : 837 
109 53! 36") 14° 37! 28” a RY FY _. 4019/94" & BY PH _ 50 4g 84/16: 8: 
a = igs 7 35! 12!\rl! == 8° 48! CRM FM! oe 19 BQ) 18 6 HY W_. 20 BO Bz 6: eee 
© 49154". BO 18! 7" “ Pu Rn JO gr 4" & Pep 10° 237 56"6 
From this it follows, that, under the assumed aie. the 
first fathom-interval is exaggerated in the ratio of 6 to 85 
whilst the others are elongated in a ratio but slightly greater 
than 6 to 8. the angle 7 (other conditions being the same 
had been taken larger, the excess of elongation of the first fath- 
om-interval would have been greater. It thus appears, that all 
of the fathom-intervals are exaggerated nearly in the ratio of 6 
to 8 or 8 to 4:—that is, sensibly in the ratio of the sines of the 
angles of incidence and refraction for water. This is only true 
in that case, cee sa may be consid lered oe ge to the an- 
bela ; the ap- 
parent Bik 2 h, sancined to be measured by the sade linear 
_ units would be 48 feet 
Strictly speaking the rays of light emanating from the succes- 
fathom marks, which, under the ee teptiem of no refrac- 
