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MATHEMATICAL CORRESPONDENCE 

QuEstion XXVII (mif-rumbered XXVI, No. XIV).—Anfwered by Hermes. 
TET x*+-2% and x7—~29:exprefs the two numbers which fulfil the two firft conditions 5 them 
if 2x71 and 4x-+1 be fquares, the queftion will be aniwered. To effect this, their re&t- 
angle 8x3207-+tarty mui be See ; fuppofe it 1 iv-fdx ? \?== 1-+-24x--25--2 ae oe 
abx3+-?x4; then, by comparing the terms, we find 224, and henee ame ; alfo 24==2— 
4==—-2, and hence /=—1;; lafitly, —4x3-++-x4-—=8%3 (fee the Append. to Hutton’s Diarian 
Mifcel. vol. iii.) ; confequently, by divifion, &c: x==12 ; which, fab{tituted in the aflumed ex- 
preflions, gives 168 and 120 for the two numbers fought. 
The fame anfwered by Mr. fames Afiton, of Harrington, E 
Put x*+2x== one of the required numbers, and «7—2x= the other ; then x?2x-11, and 
x2—2x-+1 are both evidently fquare numbers; but their fum‘is 2x*, and their difference is 4x, 
hence, by the queftion, 2x74 and 4v-+1 muft'alfo be fquares. Now put 4x--1=n2, then 
ne—IJ 1 : FE L t 
4==-~—, and here the leaft number that a will admit of (fo that 2x71 and 4x-b1 may both 
4. ; 
sil 8 : 
19 —T—s.; and hence 168 and 420 are the numbers 

_be fquare numbers) is 73 then «= 
/ 
required, to anfwer the conditions of the queftion; and the four {quare numbers are 169, 121, 
289, and 49. to 
This Queftion was alfo anfwered by Mr. Fofeph Youngs, of Norwich. 

QuEstion XXVIII (No. XIV).—Anfwered by the Rev. L. Evans, Froxfeld, Wilts. 
* By mu'tiplying the fir and fecond equations, we have y?--2xy-+-%?==2b, and the root is 
gx ,/ab. Mult. the fecond equation by xy, gives y--x—=sxy; therefore bxy==4/ab, and 


. ao B b 
gry aa ; this taken from y?+-2 xy-L-x?==ab, gives y>—2xy—+t-x?—=ab— E: > » and the root 
“is y—x—4/ es which being added and fubtraéted with y--x==,/ab, &c. gives x= 
rae ere aE UNA Y ciealiag Saas ig 
rags a/ab ja/ ub ‘ 
av ab + 1 /ub— ~, and y= 4,/ ab—Fa/ ab— — as required. 
The fame anfwercd by Mr. F. H. Hearding, of Ringfafh, Devonfhire, aged 16 years. 
The product ‘of the two given equations (x?y-Ly?x—a, and— 1. ~=#) is (Equa. 3d) x7 
x uy “ 
_axy--y?==ab; make ab=?, and we fhall have (Bqwa. 4th) x-Ly—r. . 
Again, the fecond equation multiplied by x*y?, gives x?y-L-y?x=-bx?y? ; which, taken from the 
a 
farft equation, leaves a—bx?y?==0, that is se ear 3 Make i= s*, and we fhall have (Equa. 
sth) yee | 
From the third equation fubtract 4 times the sth, and there will remain x? —2xy-by*==r2——4', 
‘Hence (Equa. 6th) »—y==4/7’—4s. Then by adding and fubtracting the 4th and 6th equa- 
, 2 een ey Pr aae 
tions, and dividing by 2, give —— ane = - 46 | 
This Ques ion was alfo ingenioufly anfwered by Mr. ¥. Afhtony Mr. T. Hickman, Mr. R. Wood; 
and Mr, ‘fof. Youngs. 
EE 
NEW MATHEMATICAL QUESTION. 
QUESTION XXXII. By. Mr. Fames Afhton, of Harrington, xear Liverpool. 
The wall of a houfe being 30 feet high, a fpout, 24 feet length, is to be fixed on the top of 
it; it is required to find the angle the fpout muft make with the plane of the wall, fo that the 
water may fall into a¥efervoir, ona horizortal plane, at ten feet dittance from the bottom of the 
wall? 

¥y* The Solutions to QUESTIONS KXV and XXVI (No. XIIT) Aawing been unfortunately 
Ici, or mifluid at the printer?sy the Authurs of thofe two Queftions are reguefted to fupply us again with 
she folutions of item, 
: ORIGINAL 
