38 Prof. Challis on Hydrodynamics. 



ferent axes may coexist even when all the terms of the series for 

 w 1 and o- 7 are taken into account. 



It is not necessary to give in detail the well-known proof of 

 the integrability of udx + vdy + wdz for small vibratory motions. 

 The proof rests exclusively on the suppositions that the velocity 

 is very small compared to #, and that no part of it is indepen- 

 dent of the time. Now these two conditions are satisfied by 

 the series for w' } if only m be very small compared to a, because 

 in that case the first term is much larger than the sum of all the 

 others. They are also satisfied by the motion at any distance 

 from the axis, since it consists of vibrations, and is of less mag- 

 nitude than that along the axis. The foregoing determination 

 of the function ^, after supposing (d ./<£ + ^%) to be equal to 

 udx + vdy + wdz, is a direct proof, to terms of the second order, 

 of the integrability of that differential for positions at any dis- 

 tance from the axis. If we suppose generally that 



{dyjr) = udx + vdy + wdz, 



we shall obtain, to small quantities of the first order, the known 

 linear equation 



dt* \& t di/ + dz*r 



which, it should be noticed, does not contain any constant of 

 specific value, such as & 2 , nor involve any assumption respecting 

 the composition of ^. Now the motion relative to an axis, 

 while obeying the laws above determined, must at the same 

 time satisfy this equation, whether or not terms involving m 2 , 

 m 3 , &c. be included, solely for the reasons that the whole motion 

 is, by hypothesis, very small compared to a, and, being vibra- 

 tory, is such as to make udx + vdy + wdz integrable. Hence, if 

 TJr l represent the value of <\|r for one such set of vibrations, we 

 shall have 



dt* V dx* ■*■ dy* "*" dz 1 )' 



So for another, 



dP ~ tt \ dx* + jfy* + d3* J' 

 &C. = &C. 



Consequently if $ = ^ 1 -f -^ 2 + ^ 3 + &c, the same equation will 

 be satisfied by the value <S> of ^, and we have also 



dx dx dx dy dy dy 



dz dz dz 



