90UD8 ON B1 BJF \< 331 



theory of oscillations round a constanl axis. Clairaul in 1735 

 leralized tin" doctrine of the simple pendulum, in an ahle 

 investigation of its conical vibrations, in which the effects ol 

 an oblique impulse were for the firsl time subjected to mathe- 

 matical determination 49 . The results for the cases in which 

 the weighl describes a circle either vertical or horizontal 

 were deduced as corollaries from the general formulas, and 

 shown to be coincident with the conclusions to which 

 Huyghens had already arrived for these simpler cases of the 

 question. A mure difficult problem still remained. When 

 a pendulous body hangs by a fixed point aboul which it maj 

 turn freely in all directions, its motion will be affected not 

 only by the obliquity of the impulse by which it is sel in 

 motion, but also by the rotation of the pendulum around the 

 line which j< >i ti> the sustaining point and the centre of gravity, 

 so thai even when this axis is dropped vertically from a state 

 of resl with the body revolving around it. this rotation will 

 sufficient, at every instant of [the motion, to wrench (as it 

 were) the axis from the direction in which it would move it 

 left at the same instant to vibrate by itself. Up to 



it time no solution of this problem has been given 

 for finite oscillations, and even for oscillations infinitely small. 

 is givi u until Lagrange published, in the firsl edition 

 of his Meeaniqw Jfautlytique, an ample dissertation on the 

 subject. After a general investigation of the free rotation of 

 ;i rigid bod}-, in which the author skilfully combines all tie 

 advantages of tin- various methods be had previously invented, 

 he proceeds to the examination of the well known case in 

 which tin- body pirouette* by virtue of tin- inertia of the ele- 

 ments alone. After a masterlj detail of all the circum- 

 stances of this case, Lagrange enters niton the discussion of 



■ 1 motions of a heavy body pirouetting aboul a fixed 

 point not the centre of gravity, and advances as far towards 



** Exam en dee diffi rentes ' '-■ illations qu'un coi 



gqu'on l<ii donne une impulsion quelconque. M m Acad. Par. II 



II. 



VOL. III. 4 P 



