38 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION 



on the change of position in the direction normal to the surface of displacement." This is 

 not true as a general proposition, hut it is true with reference to fluid motion, solely in 

 consequence of the condition expressed by the general equation (a), as appears by the reasoning 

 in Arts. 7 and S of this paper. Hence the Proposition proved in the ' Note' added to the 

 former Paper fails in giving support to the above cited assertion, because it takes no account 

 of that equation. In fact, the proof neglects the curvature of the lines of motion, and therefore 

 only amounts to shewing that in rectilinear motion a surface of displacement is a surface of 

 equal velocity when udw -¥ vdy + wdz is an exact differential, or the converse In the same 



^^ ft It fJ 13 (i ?A} 



Article (p. 378) it is said incorrectly, that 'JT'^''^ '^ J^'^V '^ '^ '^^ " °' liecause for a surface 



of displacement udx + vdy + tvdz = 0^ This is true only when the position of the surface 



of displacement through a given point is invariable, which should first have been shewn to be 



the case. The correct reasoning is given in Art. 7 of the present paper. — At the beginning 



of Art. 7 of the former paper (p. 379), it is supposed that in rectilinear motion the lines of 



motion may pass tlirough "fixed focal lines." The more complete investigation of the present 



Essay shews that they must be limited to passing through a fixed centre, or a fixed axis. — 



" dd) 

 The assertion (in p. 382) that — ^ and V are constant for a given surface of displacement 



at a given time, when udx + vdy + wds is an exact differential," is true, but, on account of 



the defects already mentioned, does not follow from any previous reasoning It is not generally 



true as asserted in p. 389, that " the variation of F at a given point is the same as if r 

 and r were constant," and consequently the equation derived from that supposition is of no 

 value. I am not aware of any other points that require adverting to. 



I proceed now to make some uses of equation (2) which will shew the importance and 

 necessity of it. 



12. First, let it be required to determine on what hypotheses the general dynamical equation 

 is intefrable. To do this it is necessary to introduce into the dynamical equation the condition 

 expressed by the equation (2), or by its equivalent equation (3). I have already gone through 

 the process for this purpose in Arts. 10 and 12 of my former paper. The result there obtained, 

 expressed in the notation of this paper, is 



It is supposed in this equation that Xdw + Fdy + Zdss is an exact differential. This condition 

 being fulfilled by the impressed forces, the equation is integrable either if the second term vanishes, 



or if JVd\// be integrable. Since — — ^= f—-ds, in the first case, -rr = and the motion 



is steady; in the other, ud^v + vdy + wdz is an exact differential. These are the only cases 

 in which the general dynamical equation is integrable. 



13. Next let it be required to find the factor — in proposed instances of motion, and 



to determine whether the motions are possible. 

 To make the equation (2), viz. 



dxb d'dr dsjy d4/ 



dt dx dy dz 



convenient for this purpose, it will be transformed into another equivalent equation in the 

 manner following. The equation \|, = 0, being by hypothesis the equation of a curve surface, 

 may be supposed to contain besides the variables x, y, z explicitly, certain parameters a, b, c, &c. 

 which are functions of the co-ordinates and the time, and vary with the varying position of a 



