318 ON THE MOTION OF 



hens', was finally in 1703 deduced by James Bernoulli, from 

 principles which have never been contested 5 . In the mean 

 time, Newton, in his Principia, had begun to calculate in cer- 

 tain cases the effects of resistance in retarding the motion of 

 points along cycloidal arcs, had reduced to the method of quad- 

 ratures, the determination of their motion along curves whose 

 planes pass through the centre of force, and had furnished 

 general principles which served afterwards to facilitate the 

 solution of the problem of the motion of a heavy point on 

 surfaces of revolution 6 . In the same work too, Newton had 

 investigated the duration of the pulses of air and the undula- 

 tions of water, and had laid the foundations of the true theory 

 of the tides 7 . Leibnitz and the elder Bernoullis had also 

 discussed with success several interesting cases of the descent 

 of a material point along given or required curves', but no 

 mathematician appears to have had regard to the form and 

 rotation of the supported mass, until John Bernoulli, late in 

 life, proposed the problem of what he called the oscillations 

 of titubating bodies 9 . In this problem none but the very 

 small oscillations are considered, and the body is supposed to 

 rock without sliding about an invariable axis, the surface of 

 support being either a plane or the concave or convex side of 

 a horizontal cylinder. After investigating the general formula, 

 Bernoulli calculates the case in which the rocking body is 

 the segment of a sphere or parabolic conoid. This rolling 



1 Hugenii Horologium Oscillatorium. Parisiis, 1673. Pars Tertia, Prop. V. 



5 Demonstration generate du Centre de Balancement ou d'Oscillation tiree de 

 la nature du Levicr. 15 Mars, 1703. — Histoire de l'Acade'mie Royalc dcs Sciences. 

 Annie MDCCIII. Paris, 1720, p. 78. — Jacobi Bernoulli opera. Geneva, 17 14. 

 Vol. ii. p. 930. 



6 Principia, Lib. II. Prop. XXV.— XXXI.— Lib. I. Prop. LIV.— Lib. 1. Prop. 

 LV. 



7 Principia, Lib. II. Prop. L.— Prop. XLVL— Lib. III. Prop. XXIV.— Prop. 

 XX XVI. XXXVII. 



8 Acta Erud. Lips. 1694, p. 276. 364. 394. Jac. Bern. Op. p. 601. 627.— 

 Leibnitii et Bern. Com. Epis. Vol. i. 23. 34. 167. 286.— Joh. Bern. Op. Vol. i. 

 120. iii. 486. 



9 De Oscillationibus Corporum titubantium super superficie aliqua immobili. 

 Job. Bern. Op. Lausanna; ct Geneva;, 1742. Vol. iv. p. 296. This paper was 

 written posterior to the year 1738. John Bernoulli was at that time 72 years of 

 age. 



