114 REPORT 1870. 



line joining the projection of any point on the plane xy to the intersection 

 of the normal with the same plane makes with the axis of (x), then ScheU- 

 bach shows (p. 300) that if S be the surface of the ellipsoid, 



S=8a6c f ^ P sin y dy d<j> 



J o J (be cos'^ sin*y+ ca sin-^ sin^y + «6 cos^y)^' 



If we put a/ - = cos T, a/ - = cos p, the expression for the surface becom os 



2x /bC^__ sinycZy 



c V a J (1— sinVcos'^y)2 (1— sin^pcos^y)! 



2rr /« Tf fiUly(?y 



c ' V 6j o (1— sinVco3Y)2 (1— sin^pcos^y)i' 



r 



If we put sinr oo?,y=<\f, siup cosy= i^lcf(x), lc= ~'^~i_ , and therefore 



Va: s c—a 



>i/ J) (I 



^■'= , , the expression for the surface may be written thus : — 



Vc — a 



^ahc I Q^^o >^b{c-a) \c-a eoQ^^o ) I 



_ Section 2. — A most interesting application of the theory of elliptic func- 

 tions to mechanics wiU be found in the 39th volume of CreUe's Journal. In 

 ■that volume is published Jacobi's memoir on the motion of a rigid body, 

 which has been already mentioned in Part II. of this Report in relation to 

 the many important discoveries it contains in pure mathematical science. 

 I now enter upon the consideration of this paper regarded as a physical 

 memoir, Jacobi makes use of the following notation. Instead of 



2K^• 2Kx 2K / 7r\ 2K/ ,r\ 



he writes 



2Xx 2Ka; 2Xx 2Kx 

 , II , e, , H, . 



TT IT TT TT 



The object of the paper is to calculate the motion of a rigid body, acted on 

 by no forces, round a fixed point. 



Let X, y, z be the fixed axes passing through the fixed point to which the 

 motion of the body is referred, the plane of x y being the invariable j)lane. 



^i' Vi' ^1 the principal axes, j>, q, r the velocities of rotation round the axes 

 otx^,y^,z,. 



the incHnation of the plane a\, y^ to the plane of x y. 



^ the angular distance of the line of intersection of these planes from the 

 axis of {x), 



<p the angular distance of the axis of a-'^ from the same line of intersection. 



ce=& x^+(3 y,+y z^. 

 y=a'x^+fi'y, + y'z^. 



z=oi'\ + B'% + y\. 



