212 New Mathematical Duefions. — { March, 
; {pz 2 R p 
Cor. 3. When tee. > AWE TBE): or tan, & i the line EB is a tangent to the cusves 
for, in this cafe, the points B,B’ correfpond, and confequently the roots of the equation are equal ; 
therefore 2Feme2 Paat-p2—=2 22 Poa (P? fec. p?-—R?), a -P?==P? fee. ¢7——R?, that is, fec. ¢—= 
LCR) 
eee we tan. ee 
Cor. 4. When Po, 2?—R*=0, or z— ER, which an{wers toa circle whofe radius==R, and 
Seentre E, 
Problem II. Fig. 2. 
To find the curve ABB’, fuch, that if, from a given point E, any right line EBB’ be dravm, 
meeting the curve in the points B, BI, a a line HG given by poftion in F, EB may be to EB’, 
as BF to B’F. 
Draw EG perpendicular to HG, and put the angle GEB==», EBor EB’==, and 27—=2Pz-+Q 
==o0, where P, Q are certain funtions of the angle 9. 
Bife& BB! in D, and ED will be == P, and BD==,/ (P?—Q) ; then, becaufe EB is to EB’, 
as BF to BF, EB/-LEB i is to BB’, as BB to B’ FFB, er ED to BD, as BD to DF ; whence, by 
5 ie PQ een 
fubftitution P:4/ (P?—-Q)::4/(P?—-Q) : DF=—,- =P— => \ 


I Pp 
: A A ' 
_ But if EG=A, EF will be aa , and DF=EBD—EF=P— eee therefore == is 
-O col. 2 
AX es AX P AxP 
Pes Oh == = and is? Po at ee oy whichis a bemexal expreffion for the curves. 
cof. cof. ' cof. © 
hav? meg the le te p' ‘Gpert ty. 
Cer. 1. Draw bK perpendicular to EG, and iet FR==x, se 9 then will 2?==x2Ly?, cof. > 
name 
cAMP fit x 7 pay? 
and arf; mae P's x7 Ya Wisin es inne 4 or ae — 2P-. 
ere id vite) x no wh CF eal 
Ne — 44% P? 

AX? 



==-0 5 whence «?--y?==4P7L. 
x Peto. 
Cor. 2. When P==R cof. 9, z7+-2Re2 cof o+A»R=o0, which fhows the curve to he a 
circle, the diftance of whofe centre C’ from Ek, and whofe radius is a mean proportional be- 
tween EC and GC. Wow, if CH and EH he drawn, as the {quare of CH is equal to the rect- 
angle ECG, and HC perpendicular to. LC, the argle CHE is a night angie, and confequently EH 
a tangent to the circle at A. Hence this theorem, which is Prop. 154. Lib. 7. of Pappus, or 
Prop. 73. De Poriimatibus. Simfon’s Opcra Poithuma: -* Circulam ABH contingant rete EH, 
EH’, .et HH* pungunturs $i dve.tur utcunque ad curculum recta = uccurratque circumferent.3¢ 
in BB’, etredie HM in F. Dico ut EB! ad EB t, ita BE ad BE.? 
7 
or x4 at x? yramAP?Px2LA A 9 Pyne A? 

Cor..3. When P= ray or z==P, the line EH wuches the curve in H for, in this cafe, 
CO <O ‘ E 
A PL. , 
Pa lab So ypy bP 6, and 2—2P 
cof. 0 f. Cxonn 
Aberdeen, Fan. 17975 fF be continued, } 

Question EXIVW (No. AL)—adafsvered by Philcmathes, of Thornbury. 

Tue true length of the pendulum, at the jaft place, muit. be Ee we ale io inches. 
TO. 6300 
And the times of ee of pendulums are dirc@ly as the fquare roots of ae ena 
1172 
ther: fore 4/39: WV 33 aac ee f 9 7,001239 fecends, the time required in which the 
“300 1$7c0 \ 
* 
pendalom will vibrate. 
This Queftion was alfo anfaered Ly By I, 

NEW MATHEMATICAL QUESTIONS. 
ore 
Question XXIX.— By 4. 2.1. X. of Oundle. 
REQUIRED to find the r2 leaft numbers, which, being divided by 2, 3, 45 §, 6, 7, 8, 9, and 
ro, refpedtively, fhall have a remainder of #3 but if divided by r1, thal! have no remainder ? 
Alva, 
