1796. ] 
feemsa little miftaken. The roots of the 
ye —h2- fh 
equation BK oy Ped, ome are ¥ 
bh 
= ay kth ekv PEED 
whofe limits of poflibility are when 4: r 
1 

23 1: 7-————==, the tangent or cotan- 
Vitaly ne 
\ gent of 150; that is, when the angle of 
the cone is 30° or 150°.-—By a fimilar pro- 
cefs, we fhall get for a maximum or mini- 
. pe ae 
OO a = X (202——y/ a —4ele-Ll)» 
whofe limits are when a:4:: 1: 
Jf 2W/3> the chord of 30° or 150°.— 
Mr. H. obferves in his fcholium, that the 
roots of the laft-mentioned equations do 
not always indicate the greateft and leaft 
feétions of which the cone is ee 
The truth is, that when the angle of the 
cone is under 30°, the plane ‘1°V in re- 
volving about T, from H towards L, 
forms a feries of elliptes whofe areas con- 
ftantly decreafe to a certain munimum, 
which is indicated by the greater value 
of v or % as given above, and then again 
increafe to a maxi/um, which 1s indicated 
by the leffer value of thefe quantities, 
again diminifhing from thence to the ver- 
tex of the cone.—This maximum, when 
the vertical angle does not exceed 23° 
54’ 20", is greater than the area of the 
circle TH, and, confequently, really the 
sreateft poffib! e ellipfe, but lefs if the 
angle is between that value and 30°, in 
which cafe, there confequently, will be 
greater ellipfes comprehended between 
the circle ITH and the abovementioned 
MiNi MUM. 
The general folution of the problem, 
in the terms in which it was ftated, I be- 
fore fent you. The queftion g giving fome 
room for {peculation with refpect to fome 
important properties of the cone, you 
will, perhaps, not deem the above remarks 
yee Mr. Hickman will not, J am 
fure, be di{pleafed at the liberty which I 
have taken with his anfwer; he appears 
a man of fcience, and is probably therefore 
not deficient in candour. 
Your's, &c. 
uly 5t4, 1796, 
J. a 
Mathematical Gis refipasleng VA 
lo the Editor lof the Montb'y Wore ie 
SIR, 
SEND anfwers tothe Mathematical 
 Queftions in your laft, and alfo to 
Queftion VII, which I de not find has 4 
yet been anfwered. I before fent an 
anfwer with - Queftion ‘<1, but the in- 
clofed is fomewhat more oe 
Yours, 6c. 
20th Fune, 1796. j. F——y. 

QuEstion VII (No. IV).—Anfwered by 
Mr. Fe F——r, 
A 
LET AB be 
the “pole,. ©. the 
eye of the obfers 
ver, DE the fur- 
face of the water, 
and FG the ex- 
tremities of the 
image of the pole, 
appearing by re- 
flection. —By the 
laws of optics, the triangles ‘AGD and CGE 
are fimilar, as are alfo the triangles BYD and 
CFE: therefore, AD : DG:: CE: GE; or AD 
ae (39): DG--GE(= 30) 2: CE (S13) : 
H==re; jand, im like aS: BD-LCE 
eae DE-+LFE(= 9O): 2, COB Cai 3): Pies 
18°5714235. And eae EG, io length of the 
image, is —FE—GE=8 5714285 feet, or & 
feet 6,, inches nearly, 
We thall alfo have (CG-LAG : cc or) DE 
: GE :: 4: 14 inch, the breadth of the image at 
the end neareft the obferver, and (CF--BY : 
CF, or) DE: FE::4: 2°47659 inches, its 
breadth at the end fartheft from him. 


Question AI (No. 
PUT » =the number of boards compofing the. 
telegraph, then the whole number of fignals 
which can be mace by it, will be ==2n——1=5 
the fum of n terms ae the following {eries, n—- 
ie —C, Sc. A, B, C, Sc. 
IiT).— Anf wered by 
Te 

ho 


say eecnniel ae 1h, 2d, 3d, &c. terms, 
or the value of the terms jimmediately preceding 
thofe in which they appear. 
If # be put equal any number of boards lefs — 
than the whole, and it be required to find how 
many fignals may be made in each of which 
_ ft boards thall be aifplayed; the “th terms of the 
foregoing feries will give the anf{wer. 

The fame anfuwer red by Mr. 7. Hickman. 
Dr. Hurtron, in his Nez es Dic- 
tienary, Vol. I, page 303, has fhown that 2 
; os 
3P2 
a 
ee 
