In Fig. 85 let the middle points of the joints be marked 1, 2, 3, etc. and the coordinates _x_ and _y_ from the crown be found for each by computation or measurement. For a load _W_ placed at one of these points, let _z_ denote the distance from it, toward the nearest skewback, to another middle point. Let ∑_zx_ be the sum of the products of all the values of _z_ by the corresponding _x_, and ∑_zy_ be the sum of all the products of _z_ by the corresponding _y_; that is, each _z_ in the last two summations is multiplied by the _x_ or _y_ of the point back of _W_ which corresponds to _z_. For a single load _W_ on the left semi-arch of Fig. 85 the following formulas are deduced from the elastic theory, _n_ being the number of parts into which the semi-arch is divided. Horizontal thrust, _H_ = (_W_⁄2)(_n_∑_zy_ − ∑_y_·∑_z_)⁄(_n_∑_y_^2 − (∑_y_)^2) (1) Moment at Crown, _M__{0} = (½_W_∑_z_ − _H_∑_y_)⁄_n_ (2) Shear at Crown, _V__{0} = (½_W_∑_zx_)⁄∑_x_^2 (3) For symmetrical loading such as _W_ on the left and _W_ on the right the horizontal thrust and crown moment due to both loads are double those found by the above formulas, while the crown shear _V__{0} is zero. For several loads unsymmetrically placed the formulas are to be applied to each in succession and the results added algebraically, the value of _V__{0} being taken as negative for the left semi-arch and positive for the right semi-arch. For any joint whose middle point is at a distance _x_ from the crown _M_ = _M__{0} + _Hy_ + _V__{0}_x_ − ∑_Wz_, _V_ = _V__{0} − ∑_W_, where ∑_W_ is the sum of all the loads between the joint and the crown and ∑_Wz_ is the sum of the moments of those loads with respect to the middle of the joint. The components of the resultant thrust normal and parallel to the joints are, _N_ = _H_ cos θ − _V_ sin θ, _F_ = _H_ sin θ + _V_ cos θ, in which θ is the angle which the plane of the joint makes with the vertical. The distances from the neutral axis to the resistance line are, at the crown, _e__{0} = _M__{0}⁄_H_, at the joint, _e_ = _M_⁄_N_. The resistance line should be located as in the vouissoir method and if not within the middle third a new design should be studied. =105. Reinforced Concrete Sewer Design.=—The method to be followed in the design of reinforced concrete arches is similar except that the moment of inertia should include both the concrete and the steel, that is, _I_ = _I_{c}_ + _nI_{s}_,