M. Foucault's Pendulum Experiments. 333 



Multiplying these equations respectively by 2dx', 2dy', 2dz', add- 

 ing thein together, and integrating, we get (since x'dx' + y'dij' 

 + z'dz' = 0), 



W + W + W =C + 2 9 z' + ^.{(z' S mX-x'co S ^ + y"}. 



Let 0= the Z POZ which the cord makes with the fixed axis. 



Then 



c 2 sin 2 6= {z' sin A-tf' cos\)* + y n . 



Hence if Y = the velocity of the ball relatively to the plane AOZ, 

 that is, the plane of the meridian, and if V ls h, and 6 l be the 

 initial values of V, z' , and 6, we have 



V 2 = V, 2 + 2£(2'-/j) + a 2 « 2 (sin 2 0-sm 2 0,)- . . («) 



Excepting for the last term, which is always very small on ac- 

 count of the factor to 2 , the expression for the velocity is the same 

 as if the earth had no rotation. 



Again, multiplying the first of the above equations by y', and 

 the second by x', and subtracting, we have 



+ <u 2 /sin \{z' cos\—a J sin X). 

 Hence by integration, 



dcd_ x , ( !l = l i_ (OCO&x{ ^ + y i*)-2<0sin\/k J %dt'} 

 dt dt y dt \. (/3) 



+ d) 2 sin \ fly's' cos\ — x'y' sin \)dt 



Putting now t** for x l2 + y n , the term — r n w cos A, is twice the 

 area described in the unit of time, by the perpendicular (? J ) from 

 the centre of the ball on the vertical OA, in consequence of a 

 horizontal angular motion of r 1 equal to w cos \. By consider- 

 ing in what directions x' and y' were reckoned positive, it will 

 appear that the constant H is positive when the motion of the 

 ball about OA is in the same direction as the earth's rotation, 

 and consequently that the negative sign above indicates that the 

 angular motion &> cos X is in the contrary direction. We have 

 thus arrived at the following general result : — 



Whatever other motion the ball may have, it has an apparent 

 motion of rotation from left to right about the vertical, equal to 

 the earth's rotation multiplied by the cosine of the co-latitude. 



It will be scon that this is a complete explanation of the fact 

 observed by M. Poncault. To compare the theoretical results 

 more closely with the circumstances of the experiment, let us 



