1 1 1- Prof. Challis on the Mathematical Theory of 



the result is 



But we have already found for points on the axis that 



And as <f> is by hypothesis independent of x and y, its value is 

 the same in this as in the preceding equation. Hence by sub- 

 stituting in the latter from the former, and striking out the com- 

 mon factor cf>, we obtain 



Since, according to this equation, /may be a function of x and 

 y only, the original assumption respecting this function and <£ is 

 proved to be legitimate. 



27. Because cr= — -^ • -jf, it appears that for given values of 



z and t, the condensation is proportional to /. By substitution 

 in the equation (8) and striking out common factors, there results 

 the equation 



d 2 cr d 2 a o 2 ct _ . . 



dx 2 dy 2 a 2 



28. Since the equation (10) is of the same form as (6), the 



process applied to the latter equation will also give a particular 



b 2 

 solution of the former. Substituting 4e for — ,-, the solution thus 



or 



obtained is 



/== a cos 2 s/ e[x cos 6 + y sin 6), 



which evidently satisfies (10), a and 9 being arbitrary constants. 

 The meaning of this result is, that the motion is wholly parallel 

 to a plane making an angle 6 with the axis of x. It must not, 

 however, be concluded that this is a general law to which motion 

 depending on the mutual action of the parts of the fluid is sub- 

 ject, simply for the reason that 6 is an arbitrary quantity. Ac- 

 cording to the principle which has all along governed the inves- 

 tigation, we must, as far as is possible, get rid of any arbitrari- 

 ness in the integral. This may readily be done by observing 

 that since the equation (10) is linear with constant coefficients, it 

 is satisfied by supposing that 



/= 2 . uh6 cos2 { */e(x cos 6 -f y sin 6) \ , 



h6 being an indefinitely small constant angle, and the summa- 

 tion being taken from = to 6 = 2tt, in order to embrace every 



