102 



in which 



and the two first of the equations (D) become 



ccz=u, + {z, + A)-'f^; ^ = ^. + (s/ + B)-^ ^ , 



if we put for abridgment 



, ^, = —1^, <P = -(t-'(3)+f/W+&C.) (I) 



z, + A ' '^'~ z, + B 



On account of the smallness of ^' ^' the quantities a , /3,, are 

 approximate values of a, /S ; and to develope a, /3, themselves, or any 

 function of them, F (a, (3), in a series of ascending powers of these 

 approximate values, we have, by the theorems of Laplace before re- 

 ferred to, 



(. [«+!]" + ' (=/ + ^)'" + ' ''" [« + 1]"+Mz/+ 5)" + ' 3 



,2 cc, «, rf«," dli.'^'Ydcc, dfiXdcc J [dfij '^dxXd^J d/3\d»J '^d^,\daj d*\di3j 5 



^"'"■^ "• ° [„ + 1]»4-1 [„' + 1]«'+1 (2;, + ^)'' + l (z, + S)"' + l ", 



the functions F„ (p,, being formed from F, (p, by changing a, ^, to 

 a,, jS,, and \ji + l]""^', [n + l]"'"^'? being known factorial symbols ; 

 we have therefore, 



da.;^' Xdd., I 



■J ^ 





[„+ i]» + i(z^4. ^)» + i 



d^d^\d»,d/3\da,) \d/i, ) ) 



y 



