TRANSACTIONS OF SECTION A. 



551 



equal to unity. The function fi was referred to above, and exhibits the increase of 

 moment of momentum, whilst the angular velocity falls. 



In the figure the dotted line shows the three principal sections of the critical 

 Jacobian and the full line shows the pear-shaped figure, the amount of departure 

 from the ellipsoid being, of course, drawn on an arbitrary scale. 



The reader who compares the figure of the critical Jacobian defined in the 

 last line of the table and shown in the figure with Poincar6's sketch will perceive 

 that it is considerably longer than he had supposed. The resemblance of the new 

 figure to a pear is also much less remarkable than in the conjectural sketch. 



OA = -65066, OB = -81498, OC ^ 1^88683 ; — = •14200 

 OM „,.„„ ON 



00 



= -75806, ^ = •78899 

 OU 



4. The Simple Pendulum without Approximation. 

 By Professor A. G. Gbeenhill, F.B.S. 



5. Sjyherical Trigonometry. 

 By Professor A. G. Greenuill, F.R.S., and C. Vernon Boys, FM.S. 



6. On the Partition of Series each Term of which is a Product oj 

 Quantics. By Major P. A. MacMahon, F.P.S. 



2 



