Engineering Mathematics, Volume I, Second Edition

Multiplying (4), (5), and (6) by x, y, and z and adding, we obtain

 

 

so,

r² + λ₁ = 0, which gives λ₁ = – r².

 

Hence, from (4), (5), and (6), we have

 

 

Putting these values of x, y, and z inn lx + my + nz = 0, we get

 

 

Since λ₂ ≠ 0, the equation giving the values of r², the squares of the length of the semi-axes, is

EXAMPLE 5.110

If a, b, and, c are positive and

 

 

show that a stationary value of u is given by

 

 

where μ is ...