602 EEPORT— 1893. 



gives 



where 

 nd 

 or 



or 

 Hence 



y J cos Q 



Vi^ ^ — and x,=:x — y — . , 



•^' smO ' ^ sin d' 



xy cos 9 

 ' ^' sin« sin2«-^' 



SP sin ' SP' sin20 ' SP ' 



2 c^ a^ cos 6 



sin (^ sin ' 



a^ cos 0— c^ sin (<=— Ci^ sin^ 0. 

 a2a;,2_2 sin2 0c, ^ rB,i/, + i,2?/, 2=^/2, 



If the axes Oa', and O//, are two conjugate diameters of tbe central 

 conic, the term x^y^ disappears in the last equation, that is to say 



c,=0. 



Thus the characteristic relation of the conjugate diameters is 



2P.-Ci2/,=0. 



In the case of rectangular axes this relation has been shown to 

 characterise the principal axes. 



It will be noticed that in the foregoing treatment distances are plotted 

 inversely proportional to the radii of gyration, and along the lines about 

 which the moments of inertia are taken. M. Levy gives another method 

 of treating the conic of inertia, corresponding to that which has been 

 already mentioned. He does this in the following manner. If to 

 any line Ov parallels KK' and KjK/ be drawn at distances from Ov equal 

 to the length of the radius of gyration relatively to this straight line, the 

 conic of inertia may be defined as the envelop of the lines K K' or K, K/. 

 Let Ou be the direction of any conjugate diameter to the line v. Let 



OA=a' 

 0B=&' 



be the lengths of the conjugate semi-diameters, corresponding to the 

 directions of Ou and Ov, and let 



6=Z.t(,0v. 



In virtue of the definition of the conic, if r is the radius of gyration of the 

 axis relatively to Ov, then 



a and h being the semi-axes of the conies ; but we also have 



ah=a'b' sin 0, 



whence 



a'b' sin , • a 



r= —— — — a sm 0, 







