42 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION 



dR *» 1 6cos0 (1 + cos^e) 1 3COS0 



It ^iU be found that ds = ^-^^ (l + 3 cos= Of, - = -^(i + 3eos^0)5 ' '""'^ 7 = 



0. 



Hence, _ + d. (^- + ^j = -p- + ^;^ • i+3,,,-..g^ 



This equation cannot be integrated independently of the equation R - e %\x^ 9 = 0; that is, it 

 can be integrated only along a line of motion. Hence the conditions of continuity are not 

 satisfied. 



Ex. 5. " Let it be supposed that the motion is in straight lines drawn from the vertex of 

 a cone and let the fluid move in parallel slices so that the motion parallel to the axis of the 

 cone is the same at all points of any section perpendicular to this axis: it is required to 

 determine whether this motion is possible. 



The equation of the surfaces of displacement is a;^ + y^ + x^ - R'^ = 0, and the equation of 



the lines of motion iZ- a? sec = 0. Hence ds = dR = dxsecG, and R = r = r. Equation (8) 



dV 2div ^, . ^ r. ., , , „, , df tdR\ 



consequently becomes — + = 0. Now puttmg / for w--\-y+e-R-, we have -^( ■— 1 



= - 2i2 F, — = 2.r, — = 2u, — = 2». Hence - ZBV + iN^x"- + y"- + :>?) -0, and N= . 



dx dy dz ZR 



Therefore u = N — = = (bM by hypothesis. Consequently v = ~ , w = ""^ , and 



da: R X 00 



V = ?^^ = d) (,r) sec 0. The above values of m, w, w, do not make itdx + vdy + ivdz an 



.17 



exact differential. Hence the dynamical equation must be integrated along a line of motion, 



dV 

 and the equation (8) with the same limitation. Consequently — = <p'(j;)sec6, and the above 



equation becomes i-i^ + - = 0, which by integration gives d)(.r) = — — . Hence V = • — -sect^, 

 ^ (p (x) X a) X' 



and the motion is completely determined. This solution agrees with the one I deduced from 



particular considerations in the Cambridge Philosophical Transactions (Vol. V. Part ii. p. 186). 



The preceding example is instructive as shewing that the motion may be rectilinear when 



uda; + vdy + wdz is not an exact differential. Another inference may also be drawn from it. 



Let the motion be steady, and let W be the velocity at a point of the axis distant by h from 



- -^ ; whence f{t) = h' W, and F = -^ 



posed to act parallel to the axis of x, the dynamical equation gives for the pressure (jo), 



j) = C - gx - -^^ sec' 0, 



W'h* 

 and if p = where x — H, C = gH ^ =^5- sec- Q. If now H be assumed to be so large that 



the second term of the expression for C may be neglected, we shall have, C = gH, and 



p = giH - X) - -^^ sec' 9. 



It might hence be argued that udx + vdy + wdz is an exact differential for this case, since C 

 is independent of sec 9. But the objection to this inference is, that if it were true, the above 

 value -of p might be differentiated supposing sec 9 variable, which would manifestly be in- 

 correct, for the result would be at variance with the differential from which this value was 

 derived. The fact is, the neglected quantity has no effect on the numerical computation of 

 p, but as it contains sec 9, we cannot regard C as independent of co-ordinates. 



the vertex. Then W=^-jf; vihence f{t) = h^ W, and V=—j-sec9. If gravity (g) he sup- 



