M. Poinsot on the Percussion of Bodies. 279 



and C from the centre of gravity G, we have the remarkable 

 equation 



S being the length of that semi-diameter of the central ellipse 

 whose direction passes through the two points in question. 

 The diameter 28 is inclined to the axis of x at an angle whose 



tangent is - ; the direction of OS is inclined to the same axis at 



X 



ax 



an angle whose tangent is — o^- (art. 1), and the product of 

 these two tangents is — -^: consequently, considering the cen- 

 tral ellipse whose equation is 



we may say that the spontaneous axis OS is always parallel to 

 the diameter 28', which is conjugate to the diameter 28, whose 

 direction passes through the centre C of the impulse given to 

 the body*. 



4. The form of the equation of our ellipse shows that /3 is the 

 length of its semi-axis in the direction of the axis of x, and « 

 that of its semi-axis in the direction of the coordinate axis of 

 y ; a. and /8 being the arms of inertia of the body with respect to 

 its principal axes GX and GY. But by forming the rectangle 



a/3= const. = R^ 

 we have, evidently, 



« = -^ and^= — , 



by which it will be seen that the two axes of the central ellipse are 

 not directly, but inversely proportional to the arms of inertia of 

 the body around the same axes. It is further manifest, from the 

 well-known formula which gives the moment of inertia around 

 any axis whatever, that this property of the two axes of the cen- 

 tral ellipse extends to all diameters of the same ; that is to say, 

 the arm of inertia of the body around any diameter is inversely 

 proportional to the length of this diameter f. 



5. If, therefore, K be the arm of inertia with respect to the 



* Salmon's 'Conic Sections,' art. 174. 



t By the formula here alluded to (see TMorie nouvelle de la rotation 

 des corps, V" partie, art. 74), the arm of inertia with respect to a ny line 

 GD making an angle -^ with the axes of x is 'Z a?co%^^-\-^^%m''^. But, 

 8 being the semi-diameter of the central ellipse whose direction coincides 

 with GD, 8 cos -^ and 8 sin ^ are the coordinates of its extremity, and as 

 such fulfil the equation of thecurve. Mcnccb-^oi-cm'^^-^- ^'^sm-^)=a?(i'^=^* , 



R- 



and the arm of inertia in question has the value -r-. 



