24 Prof. K. Clausius on the Relations between the 



count this value gives for — r- the following series : — 



J 



Ji 1 |Y , A* _!_ 2 8 -3 + ^ 2 



+A* o 8>3 5 <5 ^5 3 +&C.J. (05) 



J » - 



As the fraction -y- is equal to the fraction 4, the preceding ex- 



pression represents the ratio between the vibration- period and 

 the rotation-period. 



We might now, in order to determine J, simply divide equa- 

 tion (53) by equation (55) ; but then we should only obtain J 

 developed as far as the third power, while the term containing 

 the fourth power is obtainable ; for if we write equation (54) in 

 the form 



£( j x- j .)=4( j i> ■••:■• w> 



we can from this, on account of the factor s on the right-hand 

 side, determine to the fourth power of s the difference j 1 — Ji 

 (which does not contain a constant term, but begins at once 

 with the second power of s), although y is known only to the 

 third power. If we then form the identical equation 



O 



T ' 



and employ, on the right-hand side of it, instead of the differ- 

 ence which forms the numerator of the second term, the expres- 

 sion developed to the fourth power, we get 



J 2 s . 3 + ^ 2 4 .3 3 



-2 4 .3 4 .5 + 2 2 .3.7.29/i + 8V , . ,_„, 

 + P gio y 5 ~ ^s 4 +&c (57) 



12. We have therefore -obtained for the functions J and J 1 

 expressions from which their values can be approximately deter- 

 mined, and indeed with an accuracy proportional to the small- 

 ness of s, or inversely as the deviation of the path from the cir- 

 cular form. They afford, in many a relation, a ready insight 

 into the behaviour of these functions. 



