Prof. Thomson on Polarised Light. 201 



The interpretation of this solution, when u is taken equal to the 

 component of the earth's angular velocity round a vertical at the 

 locality, affords a full explanation of the curious phEenomena which 

 have been observed by many iu failing to repeat Foucault's admirable 

 pendulum experiment. "When the mode of suspension is perfect, we 

 have \=0 ; but in many attempts to obtain Foucault's result, there 

 has been an asymmetry in the mode of attachment of the head of 

 the cord or wire used, or there has been a slight lateral unsteadiness 

 in the bearings of the point of suspension, which has made the 

 obseiTed motion be the same as that expressed by the preceding 

 solution, where X has some small value either greater than or less 



than w, and n has the value ^ /"^. The only case, however, that 



need be considered as illustrative of the subject of the ]5 resent com- 

 munication is that in which w is very great in comparison vrith n. 

 To obtain a form of solution readily interpreted in this case, let 



[(.;- + ?i' + (X' + 4reV)4]^=w + p, [w- + rt=— (\^ + 4nV)*]*=w— ff. 



The preceding solution becomes 



a'=Acos{(w+|o)^+a} + Bcos{(w — i7)if + /3} 

 y=— Asiu {(w + p)^ + a}— B sin {(w — (t)^+/3} 



— eh. sin {(w-l-p)^ + a}-|-/B sin { {w — a)t + (i]. 



To express the result in terms of coordinates l, t), with reference 

 to fixed axes, instead of the revolving axes OX, OY, we may assume 



^:=x COS (lit— y sin w^, r] = X sin njt+y cos wt. 



Then we have 



^=Acos ipt + a,)+'B cos (fft—fi) 



+ (eA sin {(io+p)t + a.} — /B sin { (w — ff) < + /J }) sin ojt 



jj=— Asin {at + a) +B sin (at— 13) 



-l-r— eAsiu {(w-|-p)^-|-a}+/Bsin{(w — ff)^ + /3}^cosw^. 



When w is very large, e and / are both very small, and the last two 

 terms of each of these equations become very small periodic terms, 

 of very rapidly recurring periods, indicating a sliglit tremor in the 

 resultant motion. Neglecting this, and taking a=0 and /3=0, as 

 we may do without loss of generality, by properly choosing the axes 

 of reference, and the era of reckoning for the time, we have finally, 

 for an approximate solution of a suitable kind, 



$=Acosp#-|-B cos at, 



»j = — A sin p^ + B sin at. 

 The terms B, in this expression, represent a circular motion of 



period — , in the positive direction (that is, from the positive axis 

 Phil, "muq. S. 4. Vol. 13. No. 85. March 1857. P 



