44 



Hence also the time t, elapsed between any two successive 

 stages of the rotation of the body, may in various ways be 

 expressed by a definite integral ; we may, for example, write 

 generally, 



f- 



2F.m((ia)H20a)' 

 the scalar element d^ of this integral being thus expressed as 

 the quotient of a vector element, divided by another vector; 

 before finding an available expression for which scalar quotient 

 it will, however, be in general necessary to find previously the 

 geometrical manner of motion of the body, or the law of the 

 succession of the positions of that body or system in space. It 

 may also be noticed here, that the comparison of the integrals 

 (14) and (15) gives generally the relation : 



S.i'y + h^ = '2..m S\ia(^(\t. (20) 



III. When no accelerating forces are applied, or when 

 such forces balance each other, we may treat the vector ^ as 

 vanishing, in the equations of the last section of this abstract ; 

 which thus become, for the unaccelerated rotation of a solid 

 body about a fixed point, the following: 



S.7WaF. ta =7; (21) 



^.m{V.iay = -h''', (22) 



2 . 7naV. aAi = V. ly^t; (23) 



which result from (14) (15) (18), by supposing ^=0, or, more 

 generally 



^.niF.a<p = 0, (24) 



that is, by reducing the differential equation (12) of the second 

 order, for the motion of the rigid system, to the form 



S.'«^.«^ = 0. (25) 



At the same time the general relation (20) reduces itself to 

 the following : 



S.iy+h- = 0; (26) 



