of a rigid Body round ajixed Point. 1 1 



substituting for G and K their values given by (11.) and (7.)5 

 there results the equation 



'^^ = 7 1"^' ""' ' = 7-/^°t ^- dj ^"' ••• ^'^-^ 



changing the independent variable from s to z. 



d s 

 We have now to express tan 6 and -^ — , in terms of z the 



variable coordinate parallel to the real axe of the cone. For this 

 purpose assuming the equations of the ellipsoid and sphere, by 

 which the spherical conic the locus of the vertex of the axe 

 of the impressed moment is determined, 



^,+^^+^=handa:^+f+z^ = u^ .... (16.) 



a 0^ c^ ^ 



Differentiating these equations, we find 



dx _ z _^ jb^-c^) dy _ z b^ (g^-c^) , . 



dz~ X c" {a'-b'^y dz ~ y c^ {d^-b'^) ' ' ' ^ '' 



d X 



Substituting these values of the differential coefficients —r- » 



-r^« in the known expression for the differential of an arc 

 d% ^ 



of double curvature, ^^ = 1 + [jt) + (,^j ' 



and then eliminating from the resulting equation x"^ and ?/^, by 



the help of equations (16.), we find 



/ds\^_ u" {a^-c") {b'^-c^) 2;^-c^(a^-M^)(&^-M^) , 



\dz) ~ [(62-c2)^2_(-62_^2jg2J[-(^2_„2^c2_(a2_c2)z2-]l •) 



Again, to determine tan 6 in terms of ss. 



As tan^ d = -p2- = u^ ^-^ + ^T + -^f " 1» 



we find, eliminating from this expression x'^ andj/% by the aid 

 of equations (16.), 



^2 ^2 ^4 



^°^'^ = {a'^-c^) {b^-c^) u^ z2_^ (a2-«2^ {b^-u')' ' ' ^^^'^ 



ds 

 Substituting these values of-y- and of cot 6 in (15.), we obtain 



"^ ~ f A/{b'—c')z^-(,b^—u')c'V(a^—u^)cP-{a^-(^)z^'^'(^^') 



• Eliminating from (20.) the quantities a, 6, c, u, z,f, by the help of 

 equations (4.), (7.)> {'^•)y and introducing the relation h =: n^f'^, given in 

 the preceding note, we obtain 



+ yiHi.Cdr 



'^^ = [(B-C) C ;--(B A-A-)]*[(A h-k-)-{\-C) C r^f 

 the equation given by Poisson, Traite de Mecanique, torn. ii.. p. 140. 



