History of Hindu Mathematics: A Source Book v2.c17.s14.54

v2.c17.s14.54

Now, substituting the value of the factor β'² of the interpolator from the second equation, we get $Nα²β'²+k(Nα'²+k')=β²β'²,$ or $Nα²β'²+Nkα'²+kk'=β²β'².$ Again, substituting the value of k from the first equation in the second term of the left-hand side expression, we have $Nα²β'²+Nα'²(β²-Nα²)+kk'=β²β'²,$ or $N(α²β'²+α'²β²)+kk'=β²β'²+N²α²α'².$ Adding ± 2Nαβα'β' to both sides, we get $N(αβ'± α'β)²+kk'=(ββ'± Nαα')².$