124 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



the coordinates of disturbed motion. In this manner we obtain the very approximate 

 expression, 



seio 



(4 ??i2 + 7 p^j ei + 4 ei2 + 4 ;?22 4. 7 ^^ e^ ^ 4 ^2) 



y (144.) 



- 360 (^ ^3' + 7 ^73 ^3 + 4 ^32) - -2^ (p?3 + ^3) - -4032(p 



- "^ Gl' + IB ^1 ^1 + ^1^ + ^^2' + IS ^^2 ^2 + ^2') 



^^l?/" 2 j_ £i . I .2^ ^^^^gtH-^s + e^) 31vVi(^ ; 

 ~ 945 \^3 -T 16 ^3 ^3 -r ^3 y 40320 725760 



which is accordingly the sum of the terms of the fourth and sixth dimensions in the 

 development of the rigorous expression (120.), and gives, by our general method, 

 correspondingly approximate expressions for the integrals of disturbed motion, under 

 the forms 



and 



(145.) 



(146.) 



28. To illustrate by the same example the theory of gradually varying elements, 

 let us establish the following definitions, for the present disturbed motion. 



(147.) 



»1 = J?! — ^1 t, »2 = ^2 - ^2 ^? «3 = 'JS — ^^3 ^ — Y ^ *^' 

 Xj = tSTj, Xg = ^2j ^3 = ^^3 + ^ ^i 



and let us call these six quantities «i »2 *3 ^1 ^2 ^3 ^^^ varying elements of that motion, 

 by analogy to the six constant quantities e^ e^ e^ p^ p2 p^, which may, for the un- 

 disturbed motion, be represented in a similar way, namely, by (127.) and (128.), 



