IV. _The method of co-ordinates._—Given the times at which a shock arrived at five or more places, the position of which we have marked upon a map, or chart, to determine the position on the map of the centre of the shock, its depth, and the velocity of propagation. Commencing with the place which was last reached by the shock, call these places _p_, _p__{1}, _p__{2}, _p__{3}, and _p__{4}, and let the times taken to reach these places from the origin be respectively _t_, _t__{1}, _t__{2}, _t__{3}, and _t__{4}. Through _p_ draw rectangular co-ordinates, and with a scale measure the co-ordinates of _p__{1}, _p__{2}, _p__{3}, and _p__{4}, and let these respectively be _a__{1}, _b__{1}; _a__{2}, _b__{2}; _a__{3}, _b__{3}; _a__{4}, _b__{4}. Then if _x_, _y_, and _z_ be the co-ordinates of the origin of the shock, _d_, _d__{1}, _d__{2}, _d__{3}, and _d__{4}, the respective distances of _p_, _p__{1}, _p__{2}, _p__{3}, and _p__{4} from this origin, and _v_ the velocity of the shock, we have 1. _x_^2 + _y_^2 + _z_^2 = _d_^2 = _v_^2 _t_^2 2. (_a__{1} - _x_)^2 + (_b__{1} - _y_)^2 + _z_^2 = _v_^2 _t__{1}^2 3. (_a__{2} - _x_)^2 + (_b__{2} - _y_)^2 + _z_^2 = _v_^2 _t__{2}^2 4. (_a__{3} - _x_)^2 + (_b__{3} - _y_)^2 + _z_^2 = _v_^2 _t__{3}^2 5. (_a__{4} - _x_)^2 + (_b__{4} - _y_)^2 + _z_^2 = _v_^2 _t__{4}^2 Because we know the actual times at which the waves arrived at the places _p_, _p__{1}, _p__{2}, _p__{3}, _p__{4}, we know the values _t_—_t__{1}, _t_—_t__{2}, _t_—_t__{3}, _t_—_t__{4}. Call these respectively _m_, _p_, _q_, and _r_. Suppose _t_ known, then _t__{1} = _t_ - _m_ _t__{2} = _t_ - _p_ _t__{3} = _t_ - _q_ _t__{4} = _t_ - _r_. Subtracting equation No. 1 from each of the equations 2, 3, 4, and 5, we obtain, _a__{1}^2 + _b__{1}^2 - 2_a__{1} _x_ - 2_b__{1} _y_ = _v_^2 (_t__{1}^2 - _t_^2) = _v_^2 (_m_^2 - 2_t_ _m_) _a__{2}^2 + _b__{2}^2 - 2_a__{2} _x_ - 2_b__{2} _y_ = _v_^2 (_t__{2}^2 - _t_^2) = _v_^2 (_p_^2 - 2_t_ _p_) _a__{3}^3 + _b__{3}^2 - 2_a__{3} _x_ - 2_b__{3} _y_ = _v_^2 (_t__{3}^2 - _t_^2) = _v_^2 (_q_^2 - 2_t_ _q_) _a__{4}^2 + _b__{4}^2 - 2_a__{4} _x_ - 2_b__{4} _y_ = _v_^2 (_t__{4}^2 - _t_^2) = _v_^2 (_r_^2 - 2_t_ _r_) Now let _v_^2 = _u_, and 2_v_^2 _t_ = _w_. Then 1. 2_a__{1} _x_ + 2_b__{1} _y_ + _u_ _m_^2 - _n_ _m_ = _a__{1}^2 + _b__{1}^2 2. 2_a__{2} _x_ + 2_b__{2} _y_ + _u_ _p_^2 - _n_ _p_ = _a__{2}^2 + _b__{2}^2 3. 2_a__{3} _x_ + 2_b__{3} _y_ + _u_ _q_^2 - _n_ _q_ = _a__{3}^2 + _b__{3}^2 4. 2_a__{4} _x_ + 2_b__{4} _y_ + _u_ _r_^2 - _n_ _r_ = _a__{4}^2 + _b__{4}^2 We have here four simple equations containing the four unknown quantities _x_, _y_, _u_, and _w_.