<m>delim{lbrace}{matrix{2}{1}x_prime_x_-_x_0 {y prime = y - y_{0}} }}{ }</m>

Applicando le formule sopra riportate

<m>S_{x prime} = doubleint{}{}{y prime dx prime dy prime} = doubleint{}{}{(y - y_{0}) dx dy} = doubleint{}{}{y dx dy} - y_{0} doubleint{}{}{dx dy} = S_{x} - y_{0} A </m>

<m>S_{y prime} = doubleint{}{}{x prime dx prime dy prime} = doubleint{}{}{(x - x_{0}) dx dy} = doubleint{}{}{x dx dy} - x_{0} doubleint{}{}{dx dy} = S_{y} - x_{0} A </m>

<m>I_{x prime} = doubleint{}{}{y prime ^{2} dx prime dy prime} = doubleint{}{}{(y - y_{0})^{2} dx dy} = doubleint{}{}{y^{2} dx dy} - 2 y_{0} doubleint{}{}{y dx dy} + y^{2}_{0} doubleint{}{}{dx dy} = I_{x} - 2 y_{0} S_{x}+ y^{2}_{0} A</m>

<m>I_{y prime} = doubleint{}{}{x prime ^{2} dx prime dy prime} = doubleint{}{}{(x - x_{0})^{2} dx dy} = doubleint{}{}{x^{2} dx dy} - 2 x_{0} doubleint{}{}{x dx dy} + x^{2}_{0} doubleint{}{}{dx dy} = I_{y} - 2 x_{0} S_{y}+ x^{2}_{0} A</m>

<m>I_{x prime y prime} = doubleint{}{}{x prime y prime dx prime dy prime} = doubleint{}{}{(x - x_{0}) (y - y_{0})dx dy} = doubleint{}{}{x y dx dy} - y_{0} doubleint{}{}{x dx dy} - x_{0} doubleint{}{}{y dx dy} + x_{0} y_{0} doubleint{}{}{dx dy} = I_{xy} - y_{0} S_{y} - x_{0} S_{x} + x_{0} y_{0} A</m>

<m>delim{lbrace}{matrix{2}{1}x_prime_cos_alpha_x_sin_alpha_y_y_prime_-_sin_alpha_x_cos_alpha_y{ }</m>

<m>S_{x prime} = doubleint{}{}{y prime dx prime dy prime} = doubleint{}{}{(- sin alpha x + cos alpha y) dx dy} = - sin alpha doubleint{}{}{x dx dy} + cos alpha doubleint{}{}{y dx dy} = - sin alpha S_{y} + cos alpha S_{x}</m>

<m>S_{y prime} = doubleint{}{}{x prime dx prime dy prime} = doubleint{}{}{(cos alpha x + sin alpha y) dx dy} = cos alpha doubleint{}{}{x dx dy} + sin alpha doubleint{}{}{y dx dy} = cos alpha S_{y} + sin alpha S_{x}</m>

<m>I_{x prime} = doubleint{}{}{y^{2} prime dx prime dy prime} = doubleint{}{}{(- sin alpha x + cos alpha y)^{2} dx dy} = sin^{2}alpha doubleint{}{}{x^{2} dx dy} - 2 sin alpha cos alpha doubleint{}{}{x y dx dy} + cos^{2}alpha doubleint{}{}{y^{2} dx dy} = sin^{2}alpha I{y} - 2 sin alpha cos alpha I_{xy} + cos^{2}alpha I_{x}</m>

<m>I_{y prime} = doubleint{}{}{x^{2} prime dx prime dy prime} = doubleint{}{}{( cos alpha x + sin alpha y)^{2} dx dy} = cos^{2}alpha doubleint{}{}{x^{2} dx dy} + 2 sin alpha cos alpha doubleint{}{}{x y dx dy} + sin^{2}alpha doubleint{}{}{y^{2} dx dy} = cos^{2}alpha I{y} + 2 sin alpha cos alpha I_{xy} + sin^{2}alpha I_{x}</m>

<m>I_{x prime y prime} = doubleint{}{}{x prime y prime prime dx prime dy prime} = doubleint{}{}{(cos alpha x + sin alpha y)(- sin alpha x + cos alpha y) dx dy} = sin alpha cos alpha ( doubleint{}{}{y^{2} dx dy} - doubleint{}{}{x^{2} dx dy} ) + (cos ^{2} alpha - sin ^{2} alpha ) doubleint{}{}{x y dx dy} = sin alpha cos alpha (I_{x}- I{y}) + (cos ^{2} alpha - sin ^{2} alpha ) I{xy}</m>

Le ultime tre formule possono essere scritte anche nella forma

<m>I_{x prime} = {I_{x} + I_{y}}/{2} + cos (2 alpha) {I_{x} - I_{y}}/{2} - sin (2 alpha) I_{xy}</m>

<m>I_{y prime} = {I_{x} + I_{y}}/{2} - cos (2 alpha) {I_{x} - I_{y}}/{2} + sin (2 alpha) I_{xy}</m>

<m>I_{x prime y prime} = 1/2 sin(2 alpha) (I_{x} - I_{y}) + cos (2 alpha) I_{xy}</m>