14 Professor Booth on the Botation 



Integratinff this equation, 



or putting for z and t their values given by (21.) (22.), we 

 find the angle vj/ expressed by two elliptic functions, one of 

 the first, the other of the third order : 



ab c /^ d<p 



go c p "9 



a c {b'-u') 



bti ^{a^-u^) {b'—c"-) )-(27.) 



1 d^ I 



, u^ {a^-b^) . 2 -) ^l-M^sin^cf) 



>} 



b^ {a^-u^) V J 



Put N = _ i^ i^:^. 



N is the parameter of the elliptic function of the third order, 

 and it lies between —1 and — M^. 



Let« = (1+N) (l+ -^) (28.) 



Substituting for N and M their values, we find 



^2^2 (J2_y2\2 



" "^ b^u^{J-u^){y-c^y ^^^'^^ ^^ essentially positive. 



Now the parameter of the elliptic function of the thii'd 

 order, whether it shall be circular or logarithmic, depends on 

 the sign of a, being circular when « is positive, logarithmic 

 when negative; hence in this case the parameter is circular. 

 Introducing the conventional symbols for denoting elliptic 

 functions, we obtain 



~~ b^-u" F M (<}5) - -/ a n M (N, <^) (29.) 



When the time is a multiple of that in which a quadrant of the 

 spherical conic is described by the vertex of u, the elliptic 

 functions in (29.) become complete, but a complete elliptic 

 function of the third order may be represented by elliptic 

 functions of the first and second orders; hence i/r in this case 

 may be found by the help of elliptic functions of the first and 

 second orders. 



XXIII. In the particular case, where the semidiameter u 

 is equal to b, the mean semiaxis of the ellipsoid, the functions 

 by which the time t and the angle \^ are determined, be- 



