of a rigid Body round ajixed Point. 1 3 



taining the principal axes, (round which the moments of in- 

 ertia are A and B) with the plane of the impressed moment, 

 makes with a fixed line drawn in this plane at the end of the 

 time t. . , . 



Let this angle be vj/. Now it is easily shown that this angle is 

 made up of two distinct parts, one arising from the successive 

 positions which the plane of x y, or of the moments of inertia 

 A, B, assumes while enveloping a series of instantaneous 

 right cones, whose semiangles y, are given by the formula 



cos Y = — = — ^, the other arising from the uniform rota- 



' u u 



tion of the body round the axis u, with the constant angular 

 velocity ct (10.). Let the angle produced by the first cause be 

 X, then 



4t = 4^--; (25.) 



dt dt 



a little consideration will show that x ^"'l ^ ^''^ always of op- 

 posite signs. 



Now the differential of the angle on the plane of xy^ of 



which -^ is the projection, is -^ sec y ; or putting for sec 

 (Lt (It 



1 u . , dv u 



y its value — ^, It becomes -jT • T" • 



The differential of the area described by the projection of ?< 

 on the plane of xy in the time d t, is {x^ + y ) ( -jj- j ~i 



but it is also l^—r^ — ^T^V equating these values of the 



differential of the area, eliminating x, j/, -jj- , -j^ by the help 

 of equations (16.) (17.), we obtain 



-A- = =^ ^^ — 5 — - -o 5» as w—z^ = x^-\-y^. 



f 

 Hence as ct = — , by (10.) : we get 



Now it will be found that the part within the brackets is 

 essentially negative; hence changing the »ign, the formula be- 

 comes 



-4f = "{-('^)^> •••<^^-' 



