68 MOMENT OF INERTIA. [dlAT. VI. 



We call the curve represented by the above equation there- 

 fore, the curve of inertia. If we refer the curve to co-ordinate 

 axes which coincide with the conjugate diameters, the equation 

 becomes 



A 1 B' 



1 -4-1 



<** + tf ~ 



*vhere x and y are the new ordinates or tangent intercepts on 

 the axes, and A, B, the conjugate semi-axes of the curve. A 

 and B are therefore the radii of gyration of the force system, 

 measured in the direction of the co-ordinate axes, and hence 



where X and Y are the co-ordinates of the points of application 

 of the given forces. 



Since 2P#' = F SP if the sign of SP/ is the same as 2P, 

 #* is positive. That is, when all the forces act in the same 



A 8 B s 



direction %? is positive, and we have - + T = 1 which is the 



* y 

 equation of an ellipse* 



If, however, the parallel forces act in different directions, &, 

 may be positive or negative. For cases where k* is negative, 

 either A a or B a will be negative, and we shall have 



^-l!-i or ' _'+*-:FI 

 * y> ~ * y] ~ 



Both cases coincide. The double curve consists of two hyper- 

 bolas with common assymptotes, common centre, and equal 

 semi-axes. For every axis M passing through the common centre 

 O, we have a pair of parallel tangents either to one or the 

 other hyperbola. The corresponding T is positive for the one, 

 negative for the other. 



If, then, in the method of construction to which we shall 

 presently refer, the square of the semi-axis B, which lies in the 

 axis of Y, is negative, that hyperbola whose imaginary axis lies 

 in Y gives %? positive, the other gives & a negative, and reversely 

 for the other case. If the axis of moments M coincides with 



* This is readily proved. The equation of a tangent line at any point whose 

 co-ordinates are x", y\ is A s yy" + B 2 axe" = A 8 B 9 . For y = <?, we have for ' 



A 2 



the intercept on the axis of X. x = , and for x = 0, the intercept on the 



B X 



axis of Y, y = r . Substitute these values of x and y, and we obtain 



x"- v'* V 



Y? + = 1 which is the equation of an ellipse referred to its centre and axes. 



