NUMBERS. 
Prope. XIV. 
The produ& of the two formule 
ay"), is of the fame form as each o 
(stk as") (2 £ ay) = 9 hI 2? 
and, yank rege the produ& o 
of any number 
of the form (+? + 
(=: + £ ay)3 and (+!? + 
- + ays) + a (xy! 
(ea — ayy) + a(#y! 
yxy? 
of factors 
y’) will be itfelf alfo of the fame form. 
— y’s and, ibaa this latt may be transformed into 
ol former. 
we — (B41) y= (B41) (et by)?— Gat G4 1)y)* 
And 
(29+ 1) 0° — y? =((8* +1) wt by)* — (8°41) (be +3)" 
Thefe iia formule furnifh us with the following par- 
ticular on 
a —2y? = 207 — y? 
a — enya ey 
The two formule (+ z’), and (x? + z)5 wos y= ao — 9" 
are fo related to oe ‘other, a ne double of Jee pro- j 5 — y= xl — gy? 
duces the other. : ' 
x?>— 10x = 10K" — y 
5 oe ee ae ee 4 +22= fecioyaieetay 
(x+y)? + (x— yl +2 17 P= 17 xP — yl 
which is evidently of the latter form And { eyo 
2(f@+y +227) = 2+ 2y4+427= &e. &e. 
which is alfo obvioufly of the fame form as the firft. 
For example : 4= 3+ 2 +477 
2 
The produ& eara oa 2.1” 
Again : rs == 3 oe 
2 
= 30 = (3 + 2)°+ (3 — 2)? + 2” 
The produ& {= yee + 2 
That is, each of thefe forms, when doubled, produces the 
other. 
Prop. XVI. 
The formula x? — 2 y” 
another of the ferm 2 +” 
converted into the real 
is is obvious, becaufe 
s — 29 = sag — (2+ 2,9)’, and 
ast i = (wt 29) —2 (e Ly) 
as will appear from the developement te the formule ; and, 
confequently, any number that is of one of thefe rs is 
f the other. 
For example: 14 = 2.37 — 2? = 4? — 2.1? 
Alfo, 28 = 6 — 2.2%°= 2. 
may be always transformed to 
—_ ys and this laft may be again 
4 2? 
The fame transformation has place with regard to num- 
bers of the form x* — 5°; for 
#— 57 = 5 (wt 2y)— (24+ 5y)’, and 
5 — yx = (52 29)" — 5 (2e+y)?- 
Thus, in the following numbers 
age 7 — 5-2 = 5-1 — 24 = 5. 3*— 
41=5.37—-2>=19°— 5.8 = 11— 5.47 
which transformation is frequently extremely ufeful in the 
folution of Diophantine problems. 
Prov. XVII. 
If a be any number of the form 4* + 12, then will the 
formula *° — ay? be refolvable into another of the form 
Prop. XVIII. 
ra mand nx be the roots of the quadratic equation ¢? — 
P+6= a Wee will the avg of a two formule 
(2+ my) and (# + ny) be equal to wy + by’ 
This is ben from the a@ual altiplbaie of a fac- 
tors(#+my), and (x+ay). For 
(# +m) + (e+ ay) =H? + (2 +m) xy + may? 
And fince m and a are the two roots of the equation ¢* — 
«+ 6, we have, from the nature of equations, m + 2 = 
and mn = 4; and, a a the produét becomes x? i 
ax 
ees couvertsly, every wine of the form 2° + axy 
by’, may be confidered a ee lie) from the 
multiplication of two ee ee (x +a 9); 
mand n i the roots of the quadvavic! ae o* — 
Or, ah is the fame, m and a ome fuch as to anfwer 
the conditions m + 2=a, mn= 
Prop. XIX. 
Ray produd po from the multiplication of the tw 
mule x? pand x” + ax'y! + by”, is itfelf 
alfo of the fame ‘crak. 
For, by the laft, 
x +anyt by = (x + my) (x +23) 
x” + aaly! 4+ by? = (x! -+ my’) (a + 29’) 
therefore the produc in slash hk is the fame as the continued 
product of the four latter fa€tors. Now 
(# + my) ail a al ae ah Sega 
nd finc + 6=0, we have m= am— b, 
whence the eee formula becomes 
xa —byy! +m (xy + aly + ayy’) 
Or writing K= xx! — dyy! 
Yoay! + a'y + ayy') 
wehave (*+my)(c'+my) =X+mY 
foalfo, (+n y)@' +ny)=XinY 
Confequently, the whole of the above produé& is 
=(X+mY)(X4+aY)=X%*4+aXV¥4 bY 
That 
