In each case three integrals, c p c.,, c 3 , are taken, satisfying the 

 three conditions [c 2 , c 3 ]=0, [c 3 , c,]=0, [c lt e 2 ] = 0; the first being 

 the integral of vis viva, and the other two being derived from the 

 integrals expressing the conservation of areas. In the former pro- 

 blem the " principal function " is then found with great ease, and the 

 remaining integrals deduced. The set of " normal elements " thus 

 obtained coincide with those given by Jacobi (in a memoir in Crelle's 

 Journal, vol. xvii.). In the problem of rotation, the algebraical solution 

 of the three assumed integrals for y lt y 2 , y s depends upon that of an 

 equation of the fourth degree. It is therefore impracticable to 

 exhibit the principal function in an explicit form. In this respect 

 the result arrived at resembles that obtained by Mr. Cayley in a 

 totally different way ; Mr. Cayley having shown that the solution of 

 the problem is reducible to quadratures, assuming the algebraical 

 solution of a certain system of equations of the same form as those 

 to which the author of the present investigation is conducted. (Camb. 

 and Dub. Math. Journ. vol. i. p. 172.) 



Methods are then indicated by which, when one system of "normal 

 elements " is given, other systems may be found. 



The practical value of " normal solutions " of the system (I.) de- 

 pends chiefly upon the simplicity of the corresponding formulae for 

 the variation of elements, the theory of which is intended to form 

 part of the subject of the following sections. 



