136 Prof. Challis on the Mathematical Theory of 



an extensive class of problems which admit of being answered 

 by the formation and integration of ordinary differential equa- 

 tions, while at the same time the answers are not given by the 

 general integrals, but by particular solutions deducible from 

 them. These solutions contain no arbitrary constants. When 

 the answers are of this kind, they may also be obtained by 

 direct processes, independently of the general integral. In an 

 analogous manner it may be asserted, with respect to hydro- 

 dynamical problems, that those laws and relations of the velocity 

 and condensation which are independent of the arbitrary disturb- 

 ances, and are indicated by expressions which contain no arbi- 

 trary functions, may be deduced by processes which are inde- 

 pendent of the complete integral of the above-mentioned general 

 differential equation. And, conversely, any results strictly de- 

 duced from the differential equations, and admitting of inter- 

 pretation relative to the motion, but containing no arbitrary 

 functions, must be taken as exponents of laws or circumstances 

 of the motion which are not arbitrary, but depend only on the 

 mutual action of the parts of the fluid and its fundamental pro- 

 perties. These remarks maybe exemplified as follows, from what 

 has already been proved in the last communication. 



15. It was there shown (art. 5) that in all cases of fluid motion 

 we must have 



X (dyjr) = udx -f vdy + wdz, 



dt \dx 2 di/ 2 dz 2 J 



if 



Let us now assume that udx + vdy + wdz is an exact differential, 

 or, what is equivalent, that X=f('\jr, t). If we arbitrarily give a 

 specific form to the function /, the second of the above equations 

 will determine by integration the function that i/r is of x, y, z } 

 and t. Each solution thus obtained corresponds to motion sub- 

 mitted to certain arbitrary conditions ; and the above reasoning 

 proves that there may be an unlimited number of instances of 

 such motion satisfying the condition of the integrability of 

 udx + vdy + wdz. But in art. 11 of the previous communica- 

 tion, a definite result, viz. that the motion is rectilinear, is ob- 

 tained on the supposition of the integrability of that quantity, 

 without assigning any form to the function /, that is, without 

 imposing on the fluid any arbitrary conditions. That result 

 must, therefore, be interpreted with reference to the mutual 

 action of the parts of the fluid ; and the fact of its having been 

 shown to be a consequence of the integrability of udx -f- vdy + wdz 

 is in conformity with the principle enunciated in art. 9, that 

 every analytical circumstance of a general character corresponds 

 to a general characteristic of the motion, and the converse. 



