12 Professor Booth on the Rotation 



the sign + or — being taken according as z augments or di- 

 minishes. 



Now this expression is an elliptic function of the first or- 

 der, which may be reduced to the usual form by assuming 



Substituting the value of z derived from this equation in (20.) 

 we find 



abc ^ d <i> 

 t= ^ ^T, . .. . .. ., / ■ . . W ^^-.^ .(22.) 



J V (a^-i 



/ ^ (a^-«^) (Z.^ -O I 1^ [a--b^)[u- -r^) ^.^^,^ 



{a^-u^){b''-c') 



Put M^ = W 5TT70 ox ; then the modulus M varies from 



{a- — ti^){b^ — c^) 



to 1, as ?< varies from c to b, the limits of u. 



XXI. It is easy to show that the maximum and minimum 



values of s% are c- —^ ^ and c^ ~ ^^ which expres- 



sions are the squares of the vertical ordinates of the extremi- 

 ties of a quadrant of the spherical conic, and the correspond- 

 ing values of <^ are — and ; hence the time in which the 



vertex of the axis ii of the impressed moment describes a 

 quadrant of the spherical conic is given by a complete elliptic 

 function of the first order. Calling this time T, we find 



T ^^^ f ^ ^^ • .(23) 



/ ^/ (a2_2^2) (J2_c2) / ^ 1 _ M2 sin2cf>' " ^ '' 



hence we may express z in terms of t, let s = F [t). Substi- 

 tuting this value of ~, in the simultaneous equations (16.) we 

 obtain the equations 



x = F(0, 2/=F''(0, ^= F(0. . . . (24.) 

 By the help of the last of these three equations we can deter- 

 mine the angle which the plane of the principal axes x,y 

 makes with the axis u of the impressed moment at the end of 

 the time t ; and from the two former we find the angle which 

 the projection of ?i, on the plane of ocy, then makes with the 



F " (t) 

 principal axis x ; for the tangent of this angle = -^UA' 



XXII. We must now, in order to the complete determina- 

 tion of the position of the body at the end of the time /, de - 

 termine the angle which the intersection of the plane cou- 



