OPTICS. 
593 
by a lens of proper fize into a focus, which was moved 
flowly along the needle, beginning at the middle, towards 
one of the extremities, and always towards the fame ; 
taking care never to move it back in a contrary direction. 
After this operation had been continued half an hour, 
the needle was examined, and found not to have ac¬ 
quired any perceptible polarity or attraction. The pro- 
cefs was refumed for 25 minutes more, making in the 
whole 55 minutes, at the end of which time the needle 
was found to be ftrongly magnetic ; the point on which 
the violet ray had been moved flying from the north pole, 
and the, whole needle powerfully attracting iron filings. 
Dr. Carpi a flu red the profeffor that a clear and dry atmo- 
fphere is eflential to the fuccefs of the procefs ; but that 
the temperature is a matter of indifference. New Monthly 
Blag. Dec. 1817. 
Of the RAINBOW. Plate VII. 
Introductory Propositions. 
,Prop. I. If two quantities bear an invariable ratio to each 
other, their correfponding increments are in the fame 
ratio. 
Let X and Y be two quantities, x and y their corre¬ 
fponding increments. Then, by the fuppofition, X : Y 
1: X -f x : Y -f - y; and alternately, X : X + .r :: Y : 
Y -p y; by divifion, X : x :: Y : y; therefore, alter¬ 
nately, X : Y :: x : y. Hence the increment of nx is 
n times the increment of x. 
Prop. II. If the fines of two arcs be always in a given 
ratio, the evanefcent increments of the arcs are propor¬ 
tional to the tangents of thofe arcs. 
Let AB (fig. 1.) be a circular arc, whofe radius is 
CA, fine BD, and tangent AT ; draw EG parallel and 
indefinitely near to BD, BF parallel to DG, and join 
EB. Then, the triangle CBD is fimilar to the triangle 
EBF, formed by EF, FB, and the chord BE; for, the 
angles CDB, BFE, are right angles ; and the /_ EBC is 
a right angle, (Newt. Princip. Lem. 6.) and therefore 
equal to the /_ FBD ; take away the common angle 
FBC, and the remaining angles, CBD and FBE, are 
equal. Hence, FE : BE :: CD : CB ; and, in the fimi¬ 
lar triangles CDB, CAT, CD : CA (CB) DB : AT; 
therefore, FE : BE :: DB : AT; whence BE— 
and BE is ultimately equal to the increment of the arc 
AB^ (Princip. Lem. 7.) confequently, BE, the increment 
FE X AT 
of the arc,2=—-; and fince FE, the increment 
of the fine, varies as DB, the fine (Prop. I.) BE varies 
as AT. 
Prop. III. If a ray of light, refrafled into a Inhere, 
emerge from it after any given number of reflections, 
to determine the deviation of the ray, and the angle 
contained between the direftions in which it is incident 
and emergent. 
Let a ray of light RA, (fig. a.) incident upon the 
fphere ABCD at A, be refracted in the direftion AB ; at 
B let it be reflected in the direction BC ; and at C in the 
direction CD : at D let it be refrafted out of the fphere, 
in the direriion DR ; produce RA, RD, to M, Q. Take 
O the centre of the fphere ; join OA, OB, OC, OD ; and 
let A be the angle of incidence of the ray R A ; B the an¬ 
gle of refraction ; R, a rigjit angle. Then the /_ OAM 
—A , the OAB—the z OBA (Euc. 5. 1.) 222 the z 
OBC = the Z OCB = the Z OCD = the /ODC=B. 
Alfo, the angles of deviation at A and D are equal; for, 
if BA be fuppofed to be incident at A, the angle of inci¬ 
dence BAO is equal to the angle of incidence CDO of 
the ray CD ; therefore the angles of deviation are equal; 
and, fince the angle of deviation at A is A—B, the whole 
deviation arifing from the two refra&ions is 2A — 2B. 
Voh. XVII. No, 1200, 
Again, the angle of deviation at B is 2R—2B, and the 
angle of deviation at every other reflection is the fame ; 
therefore,; if there be p reflections, the whole deviation, 
arifing from this caufe, is z/iR— 2pB. To this, let the 
deviation arifing from the refractions be added, and the 
whole deviation of the ray from its original direction is 
2/1R — 2 . p + 1 . B -}- 2 A. 
Alfo, a deviation through the angle 2 pR, which is a 
multiple of 180 0 , produces no inclination of the emer¬ 
gent to the incident ray; therefore, the inclination is 
reprefen ted by 2 A—2 . p + 1. B ; or 2 .p -f 1 . B—2 A. 
Prop. IV. If a fmall pencil of parallel homogeneal rays 
be refrafted into a fphere, and the ratio of the fine of 
incidence to the fine of refraction be known, to fine at 
what angle the rays muft be incident, that they may 
emerge parallel after any given number of reflections 
within the fphere. 
Let RAM, ram, (fig. 3.) be the directions of the inci¬ 
dent, DN, r In, the directions of the emergent, rays; pro¬ 
duce ND, ml, if neceflary, till they meet RM, rm, in M 
and m. Then, fince AM, am, as alfo DN, dn, are paral¬ 
lel by the fuppofition, the angles at M and m are equal ; 
therefore, when the rays are incident at, or near to, A, 
the angle RMN, contained between the incident and 
emergent ray, ceales to increafe, or decreafe; and there¬ 
fore, the notation in the laffc article being retained, 2.p-fi 
. B — aA, and confequently p-\- x -B—A, ceafes to in¬ 
creafe or decreafe ; that is, the increment of p -}- 1 . B, is 
equal to the correfponding increment of A. Alfo, fince 
fin. A is in a given ratio to fin. B, The increment of B 
: the increment of A :: tang. B : tang. A (Prop. II.) 
or, multiplying the firfl: and third terms by p -f- 1, p + 1 
X increment op B y, increment of A p + 1 . tang. B : 
tang A ; and p -J- 1 X increment of B = increment of 
p + 1 . B, (Prop. I.) Therefore, the increment of 
P + 1 .B : the increment of A :: p + 1. tang. B : tang. 
A; and, fince the increment of p + 1 .B is equal to the 
increment of A, when the rays emerge parallel, p -f- 1 
.tang. B = tang. A; or, tang. A : tang. B :: p +i : 1. 
To determine the angles A and B, f'uppofe x and y to 
be their colines, the radius being unity; and let fin. A 
: fin. B :: m : n. 
VT-^x 2 
Then, 3/1— a 2 = fin. A ; 3/1— y 2 — fin. B ; 
— tan g- A; ^ ty- — ~ tang. B. And, from the rela¬ 
tion of the required angles, we have the following pro- 
portions; 
V'—n 2 
• •• p+ 1 : s 
and 3/1— y 2 
1 
by -compofition, - 
3/1- 
1 
V 
m; 
P+ 1 
hence, y =2 
+ 1 • nx ; and y 2 = P±_ 
; therefore, 1— y 2 
, 112X 2 
; and, by 
multiplying extremes and mea ns, n 2 — n 2 x 
H — 1 . n 2 x 2 — in 2 
i 2 x 2 ; hence, 71+1! 
■n“ ; or 
: x :: 
p+1] 2 .n*x 
3/ p 2 +2p . nx= 3 / m 2 —u 2 ; confequently, 1 
y jP 2 p . n : \/ m 2 — n 2 . The cofine of A being 
determined by this proportion, the angle itfelf inay be 
found from the tables. Alfo, m : n :: fin. A : fin. B ; 
and, the three firft terms in the proportion being known, 
the fourth is known; that is, fin. B is known; and 
therefore the angle B may alfo be found from the tables. 
7 M The 
