Encyclopaedia Britannica, 11th Edition, "Constantine Pavlovich" to "Convention" / Volume 7, Slice 2

by Various

we have the relations p_n = a_np_n-1 - b_np_n-2, q_n= a_nq_n-1 - b_nq_n-2.

Taking the quantities a₁ ..., b₂ ... to be all positive, a continued fraction of the form

a1 +	b2		b3		...,
	a2	+	a3	+

is called a continued fraction of the first class; a continued fraction of the form

b2		b3		b4		...
a2	-	a3	-	a4	-

is called a continued fraction of the second class.

A continued fraction of the form

a1 +	1		1		1		...,
	a2	+	a3	+	a4	+

where a₁, a₂, a₃, a₄ ... are all positive integers, is called a simple continued fraction. In the case of this fraction a₁, a₂, a₃, a₄ ... are called the successive partial quotients. It is evident that, in this case,

p₁, p₂, p₃ ..., q₁, q₂, q₃ ...,