350 Prof. Challis on the Motion of a small Sphere 



the above expression for the part of cr f which does not change 

 sign will become 



the constant 6 being in general different for every different value 

 of X. It may here be remarked that, when the motion is com- 

 pounded of simple vibrations having the same value of X, the 

 composite velocity to the first order of small quantities has, 

 according to the foregoing reasoning, the value 



2tt 

 wnf 'sin — (z — dt + 6) ; 



and consequently, although m is an absolute constant, there may 

 be vibrations of this type of all degrees of magnitude, the maxi- 

 mum velocities depending on different values of n. 



Having thus determined how far the law of coexistence applies 

 to vibrations relative to a single axis, we may proceed to consider 

 similarly the coexistence of vibrations relative to different axes 

 having any positions in space. Hitherto the axis of z has been 

 supposed to coincide with the axis of the vibrations. Let us now 

 conceive the origin and the directions of the rectangular coordi- 

 nates to be any whatever, and let the coordinates of the previous 

 investigation be transformed so as to be referred to the new 

 origin and axes. After the transformation we shall still have 

 udx 4- vdy + wdz an exact differential, because this condition does 

 not depend on the position of the axes of coordinates, but on the 

 character of the motion. Hence if ^ T represent the transformed 

 value of t|/-, we shall also have, to the first order of small 

 quantities, 



df- '\dx* + dy* + dz* V 



Similarly, if \jr^ be the transformed value of i|r for the vibra- 

 tions relative to a different axis, we should have another such 

 equation with tJt 2 in the place of yfr v and so on to any number 

 of sets of vibrations. Now since, from what is proved above, 

 these different vibrations are such as may exist separately, it fol- 

 lows, from the equations being linear with constant coefficients, 

 that they may exist conjointly. For if the equations be added, 

 and yjr be put for ^ 1 + ^ 2 -{-t|'\ j + &c., it will be seen that the 

 sum of the particular values yfr v ^r^, &c. satisfies the same equa- 

 tion that they satisfy separately. After obtaining by this first 

 approximation a composite value of ^ of explicit form, we might 

 proceed to a second approximation by substituting this value in 

 the terms of the equation (C) which are of the second order, it 

 being still supposed that (dyjr)=udx-\-vdy + wdz. But from the 



