[ 287 ] 



XLI. Remarks on a Statement in the Traite de Mecanique 

 o/Toisson. By James Booth, M.A., Principal of, and 

 Professor of Mathematics in Bristol College*. 



T N a work of great and deserved celebrity, to be found in the 

 -*- hands of every student of mechanical science — the Traite de 

 Mecanique of Poisson— the author, in determining the three 

 principal axes of rotation of a body, states correctly, that a 

 principal axe passing through the centre of gravity is also a 

 principal axe for any point assumed along this line, but seems 

 to have committed an oversight in saying that the two re- 

 maining principal axes of the assumed point will vary in di- 

 rection as the point shifts along the line. His words are, after 

 giving the formula for determining the angle between the 

 principal axe Ox 1 of the point O, and the axis of coordinates 

 O x, " Les integrates que cette equation renferme pourront 

 changer de valeur avec la position du point O ; en sorte que 

 le long de l'axe O z, les deux autres axes principaux ne seront 

 pas, en general, paralleles a eux-memes." — Traite de Meca- 

 nique, torn. ii. page 91. 



The equation by which is determined, as given by Pois- 

 son, is 



(cos 2 0— sin 2 6) fxy d m + sin cos {fx* dm—y^dm) — 0, 

 which may be reduced to 



t an 20 = Ifxy*™ 



y*(j/ 2 — # 2 ) dm ' 

 Now of the three coordinates x y z of the particle d m, the 

 only one which varies by shifting the point O along the axis 

 of Z is z; but the above expression for the value of tan 2 is 

 independent of z, hence the angle is constant, or the prin- 

 cipal axes are always parallel for any point along the line. 



A direct demonstration from established principles of the 

 parallelism of the principal axes in this case may be given as 

 follows : — 



Assuming then that a principal axe of rotation through the 

 centre of gravity, is also a principal axe for any point taken on 

 this line, let G X, G Y, G Z be the three principal axes of the 

 body passing through the centre of gravity G, O any point as- 

 sumed on the axis G Z ; if the three principal axes through 

 this point are not parallel to those, through the centre of 

 gravity, let the principal axe O x' make with the line O x, 

 which is parallel to the line G X, the angle w ; let the mo- 

 ments of inertia round the principal axes G X, G Y, G Z be 



* Communicated by the Author. 



