liv 

 if we make for abridgment 



= i< 



^'j^ + s'^y (9) 



and denote by ^oq and -p the values which the variation of a, 



and the differential coefficient of that vector taken with respect 

 to t, are supposed to have at the origin of the time. The defi- 

 nite integral denoted here by the letter f is the same which 

 was denoted by the letter s in the Essays already referred to, 

 and which was called, in one of those Essays, the Principal 

 Function of the motion of a system of bodies ; and if we now 

 regard it as a function of the time t, and of all the final and 

 initial vectors, a, a', . . a^, a'o, . . of the various bodies of the 

 system, and suppose (as we may) that its variation, taken 

 with respect to all those vectors, is determined by an equation 

 of the form, 



= 2Sf 4- S(aSa - (T^^a^ + Sa . tr - Sao . tro), (10) 



in which a, (Tq are vectors, we are conducted, by comparison 

 of the coefficients of the arbitrary variations of vectors, in the 

 equations (8) and (10), to the two following systems of for- 

 mulae : 



da ,da' , . 



^^ = -^, m'^=.'o,...; (12) 



of which the former may be regarded as intermtediate, and the 

 latter as final integrals of the differential equations of motion. 

 The determination of the (vector) coefficient cr, from the varia- 

 tion of the (scalar) function f, is an operation of the same 

 kind as the known operation of taking a partial differential co- 

 efficient, and may, in these new calculations, be called by the 

 same name ; but in order to be fully understood, it requires 

 some new considerations, of which the account must be post- 

 poned to anotlier occasion. 



