314 Prof. Challis on the Motion of a small Sphere 



three ; whereas in almost all the physical questions to which 

 analysis has hitherto been applied, methods of solution have been 

 employed which involve differential equations between thevariables 

 such that they are reducible to equations each containing no more 

 than two variables. On account of the comparatively few applica- 

 tions that have been made of partial differential equations, I have 

 derived little assistance in my hydrodynamical researches from an- 

 tecedent or contemporary mathematicians,andhave been compelled 

 to attempt the discovery of new processes. When in doing so 

 I have ascertained that an adopted course of reasoning has proved 

 fallacious, I have not hesitated to indicate the reason of the 

 failure, considering that every such indication tends to clear the 

 way for finding the true course. Instances of such rectifications 

 of previous methods will occur in the present communication, 

 the special object of which is to point out the course of analytical 

 reasoning which is alone compatible with the assumed properties 

 of the fluid and the given conditions of the problem, and to ex- 

 hibit as succinctly and connectedly as possible the particular pro- 

 cesses required for its solution. What is here attempted is a mere 

 matter of reasoning from admitted premises : in another com- 

 munication I shall endeavour to show that the results to which 

 the reasoning leads are absolutely necessary for the progress of 

 theoretical physics. 



In the course of the Supplementary Researches (Parts I., II., 

 and III.) contained in the Philosophical Magazine for September 

 and October 1865, and January 1866, certain conclusions are 

 drawn which it will be convenient to cite here, as they bear 

 essentially on the present investigation. First, the expression 

 for the velocity (w) along an axis about which the condensation 

 and transverse vibration are symmetrically disposed having been 

 obtained, and the relation between that velocity and the conden- 

 sation (<r), to terms of the second order, being found to be 



w 

 aa=/cw-}- (/e 2 — 1) — > 

 v la 



it was inferred, the motion being wholly vibratory, that the con- 

 densation corresponding to the second term of this equation has 

 the effect of making the excursion of a given particle, when in 

 the condensed half of a wave, equal to its excursion when in the 

 rarefied half. 



Secondly, reasons were given for concluding that, although the 

 series for w and a were derived from equations that are not 

 linear, the velocities and condensations expressed by different 

 sets of such series may coexist. As, however, it appeared from 

 further consideration that the reasons alleged were not valid to 

 the extent I supposed, I shall now enter upon a new invest iga- 



