Prof. Challis on the Hydrodynamical Theory of Vibrations. 297 



ively to the resolved parts of the original transverse motion in 

 the same directions. That is, if the condensation a at any point 

 of the axis be separated into cr j and cr 2 . and if/, f\,f. 2 refer re- 

 spectively to the original and the bifurcated motions, we shall 

 have the four equations 



<T = (T i + CT 2 , 0"/= (T l /, + 0\ 2 / 2 , 



4Ti dfc, df x df, , #1 <//" ?/ 



ax * ax dr r • dy cly clr r 



by which «r„ cr.^fvfz are determined. 



40. At the end of the communication of December 185.2, I 

 have obtained a particular solution of the equation 



&f ^f a n n 



d+a4 + ^=° 



by a process analogous to that which was applied to the equa- 

 tion (6) ; and as the particular solution of the latter indicated 

 laws of the motion independent of arbitrary disturbances, that of 

 the other must receive a like interpretation. The solution in 

 question is 



/= cos {2 s/~e[ x cos 6 + ?/sin 6) ) , 



the value of/ being supposed to be unity where x = Q and y = 

 and 6 being an arbitrary constant. All that can be generally 

 inferred from this value of/ is, that independently of the parti- 

 cular mode of disturbance, the motion may be parallel to an 

 arbitrary direction. Let us, therefore, now suppose that the 

 original motion is separated into two motions in fixed directions 

 corresponding to the angles 6 i and 2 . Then, by the reasoning 

 of the preceding article, 



[<?/= <?\f\ + 0-2/2 = °"i cos { 2 ^ e (* cos #i + V sin #i) \ ' 



+ o- 2 co8 { 2 */e [x cos 6 2 + y sin 2 ) \ , 



which value of / evidently satisfies the differential equation. 

 By expanding to the second powers of x and y, we have 



o/=cr l -f o" 2 — 2(7 1 e(o, ,2 cos 2 9 x + 2xy sin9 i cos 1 + y'sm <2 X ) 



— 2<r 2 e(a? 2 cos 2 2 + 2<ry sin # 2 cos # 2 + ij 2 sin 2 # 2 ) . 



But from the considerations adduced above, this value of of does 

 not apply to any actual motion unless it be identical with that 

 for free motion, which, to the same degree of approximation, is 

 c(l— er). This condition involves the following consequences : 



a 



# 2 = g + B i3 tr x + (7 2 = g- } cr i = o\ 2 = ^ ; 



that is, the two component motions are at right angles to each 

 other, and of equal magnitude. 



Phil, Mag, S. 4. Vol. 24. No. 161. Oct. 1862. X 



