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1 
XX Vi. A new Method of finding the equal Roots of an Equation, by 
Divifion. By the Rev. John Hellins, Curate of Con- 
dantine, in Cornwall ; communicated by Nevil Malkelyne, 
D. D. F. R. S. and AJlronomer RoyaL 
Read June 20, 1782. 
1 
rr^HE following theorems are a produdion of juvenile 
I years. They were invented about twelve years ago, 
when algebra was my favourite dudv ; and one of them (the 
find) was publidied as a fpecimen of this method of extrading 
the equal roots of an equation about ten years ago. Since that 
time my avocations have left me but very little leifure for im- 
proving any invention of this kind. Thefe theorems, then, 
are in their crude date ; however, fuch as they are, I flattei 
myfelf, they will adord an ealier folution of equations that have 
equal roots than is generally known, and be acceptable to the 
ingenious algebraid. 
theorem I. 
If the cubic equation x 3 -px % + qx-r = o has two equal roots, 
each of them wilt he (x) = —g-- ■ 
r> E M o N S T R A T I O N. 
Call the three roots a , a , and b\ then, by tne composition 
of equations we (hall have aab = o, where 
Vol. LXXI-I. 1 ‘ > . 2 * 
